Piketty v. Solow

Krusell and Smith lay out the Solow and Piketty growth models very nicely but perhaps not in a way that is immediately transparent if you are not already familiar with growth models. Thus, in this note I want to lay out the differences using the Super Simple Solow model that Tyler and I developed in our textbook. The Super Simple Solow model has no labor growth and no technological growth. Investment, I, is equal to a constant fraction of output, Y, written I=sY.

Capital depreciates–machines break, tools rust, roads develop potholes. We write D(epreciation)=dK where d is the rate of depreciation and K is the capital stock.

Now the model is very simple. If I>D then capital accumulates and the economy grows. If I<D then the economy shrinks. Steady state is when I=D, i.e. when we are investing just enough each period to repair and maintain the existing capital stock.

Steady state is thus when sY=dK so we can solve for the steady state ratio of capital to output as K/Y=s/d. I told you it was simple.

Now let’s go to Piketty’s model which defines output and savings in a non-standard way (net of depreciation) but when written in the standard way Piketty’s saving assumption is that I=dK + s(Y-dK). What this means is that people look around and they see a bunch of potholes and before consuming or doing anything else they fill the potholes, that’s dK. (If you have driven around the United States recently you may already be questioning Piketty’s assumption.) After the potholes have been filled people save in addition a constant proportion of the remaining output, s(Y-dk), where s is now the Piketty savings rate.

Steady state is found exactly as before, when I=D, i.e. dK+s(Y-dK)=dK or sY=sdK which gives us the steady level of capital to output of K/Y=s/(s d).

Now we have two similar looking expressions for K/Y, namely s/d for Solow and s/(s d) for Piketty. We can’t yet test which is correct because nothing requires that the two savings rates be the same. To get further suppose that we now allow Y to grow at rate g holding K constant, that is over time because of better technology we get more Y per unit of K. Since Y will be larger the intuition is that the equilibrium K/Y ratio will be lower, holding all else the same. And indeed when you run through the math (hand waving here) you get expressions for the Solow and Piketty K/Y ratios of s/(g+d) and s/(g+sd) respectively, i.e. a simple addition of g to the denominator in both cases (again bear in mind that the two s’s are different.)

We can now see what the models predict when g changes–this is a key question because Piketty argues that a fall in g (which he predicts) will greatly increase K/Y. Here is a table showing how K/Y changes with g in the two models. I assume for both models that d=.05, for Solow I have assumed s=.3 and for Piketty I have calibrated so that the two models produce the same K/Y ratio of 3.75 when g=.03 this gives us a Piketty s=.138.


As g falls Piketty predicts a much bigger increase in the K/Y ratio than does Solow. In Piketty’s model as g falls from .03 to .01 the capital to output ratio more than doubles! In the Solow model, in contrast, the capital to output ratio increases by only a third. Remember that in Piketty it’s the higher capital stock plus a more or less constant r that generates the massive increase in income inequality from capital that he is predicting. Thus, the savings assumption is critical.

I’ve already suggested one reason why Piketty’s saving assumption seems too strong–Piketty’s assumption amounts to a very strong belief that we will always replace depreciating capital first. Another way to see this is to ask where does the extra capital come from in the Piketty model compared to Solow? Well the flip side is that Solow predicts more consumption than Piketty does. In fact, as g falls in the Piketty model so does the consumption to output ratio. In short, to get Piketty’s behavior in the Solow model we would need the Solow savings rate to increase as growth falls.

Krusell and Smith take this analysis a few steps further by showing that Piketty’s assumptions about s are not consistent with standard maximizing behavior (i.e. in a model in which s is allowed to vary to maximize utility) nor do they appear consistent with US data over the last 50 years. Neither test is definitive but both indicate that to accept the Piketty model you have to abandon Solow and place some pretty big bets on a non-standard assumption about savings behavior.


"Piketty" is an example of Macroeconomics and is therefore wrong.

If growth is declining, holding labor skill levels constant, which do you think is more likely: an increase in capital investment and capital intensity (new technological investments and replacement of depreciated equipment) OR programs to retrain labor and increase human capital.

Why do you assume labor needs to be retrained? That implies we have structural issues. We don't. We are still in an AD hole.

I am responding to the Piketty critique, which is a long run prediction.

Besides, we can have both structural issues and a current AD hole.

Neither. Look at buildings in slow growth areas. They are not maintained, their values decrease. There are lots of qualified people able and willing to work to keep them up, but it isn't done.

Your super Solow model seems pretty dumb.

Labor growth is a pretty big deal, as is capacity utilization.

We have really low growth in part because a big chunk of economic resources are simply noe employed.

When all our resources are employed, we actually get better technology growth because everyone is trying to find a way to save resources.

So again, your model is dumb.

He's simplifying things so that non-econ people can understand it. I teach this model in intro macro, using this textbook, at a fairly elite US college, and it's not found trivial (hence the simplification). The grown-up models have labor growth, etc.

Your criticism is dumb.

My guess is Nick Bradley does not comprehend multi-variable calculus. Otherwise he'd realize you will get similar results letting labor vary as you will holding it constant.

You could let labor vary, but then you are going to need multi-variable calculus. You are going to lose a lot of liberals if you invoke partial derivitives. Too complex for their underdeveloped brains.

I have found no discernible correlation between student political ideology and comfort level with multi-variable calculus. I have, however, found that the dogmatic (liberals/conservatives/libertarians/misc, doesn't matter who) tend to resist complex models.

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A textbook that costs only $199.99 from the publisher?

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'If you have driven around the United States recently you may already be questioning Piketty’s assumption.'

If you have driven around Germany, Austria, Switzerland, or France (OK, the route Patton took, more or less, from St. Malo to the German border) recently, you wouldn't.

But then, one can always rely on Americans to think that only the U.S. matters, even when discussing a French citizen, writing in French, living in France, teaching in France, and the director of the French École des hautes études en sciences sociales.

But didn´t Piketty make assumptions about US? Didn´t him use US data? If he would like to let it concern only to France them he should have used only french data, no?

Most of his data was European-- his most detailed was.

Look, you clearly have not read his book. Clearly.

Once again, Krusell and Smith have a deprecation rate of 10%/year, which is much much too high. You shouldn't be citing them favorably...

As I have been playing around with this, I have found that it is not depreciation but rather the dependence of the (gross of depreciation) marginal product of capital on the capital-output ratio that is the greater attenuator in neoclassical models...

I use a 5% rate but 10% doesn't seem much, much, too high. Here are rates from Jorgenson and Kun


Who is doing your algebra for you, Broad DeLarge? We all know you can't manage even simple logical arguments on your own.

It has to really bug you that Lee Ohanian is so much better at everything than you are, including economic history.

As the economy shifts from steel production type technology to silicon/information technology, depreciation shifts to high %'s /yr as we have moved past the old physical rust out model of capital asset depreciation to becoming obsolete and having the capital value go to zero as new silicon does ten times more (performance/cost) in ten years. Even human capital in terms of knowledge can become obsolete for anyone who doesn't spend a great deal of effort and time in staying current.

A computer scientist who doesn't stay current is obsolete is a few years. I know this well as when my son went to Berkeley and just finished his first journal article, I asked him what is was about and he said, "dad, it will take me three hours go give you the vocabulary to read the abstract" and I downloaded it and he was right. The whole meaning of "capital" relative to "knowhow" is changing under our feet.

At a time when Moore's Law is still valid and scientific knowledge is growing at exponential rates combined with that fact that any new innovation can interact with all existing innovations creating new opportunities for innovation at an N! type rate (opportunities growing even faster than an exponential function), it seems that any set of assumptions holding technology constant and assuming a decreasing g may not be even close to representing reality of the world.

In my little corner of the universe, existing forces and groups are desperately trying to prevent innovation from disrupting their comfortable existence. For example, the commercial fishermen running de-marketing propaganda efforts against aquaculture (with tax payer money), as aquaculture is growing at 9%/yr world wide outside the US, where regulatory agencies, responding to the propaganda, are forcing tens of thousands of jobs to go outside the US.

In countries where the existing entrenched interest are able to use government power to freeze the technology evolution and achieve zero population growth, perhaps the assumption of no technological change and lower growth may be realistic. If obsolesce can't destroy capital value and a monster 3D printer doesn't make cheap, perfect insulated houses in a day, the existing politically powerful and connected rent seekers will become even more unequal.

Another funny thing about the Piketty model - in steady state where g=0 you get that I = D = dK = Y, that is to say, the economy's entire production is used to replace depreciating capital. Reductio ad absurdum?

g is never zero in this model since r is constant and there is no growth due to changes population or technology then the change in capital K would be sY and the change in income would be rsY where s is the conventional savings rate minus depreciation so the ratio never becomes greater than 1/r

Piketty's accuracy is entirely irrelevant. He's a Smart Economist (and European, too -- it's a Scientific Fact that foreigners are smarter than Americans) who says that we should take stuff away from Rich People. That's all that matters. The media will tout him as the Greatest Economist Ever and anyone who disagrees with his conclusions is a denialist and a dumb American and probably racist, to boot.

Judging Piketty in terms of actual economics is like judging Barack Obama in terms of actual governance. Those standards simply don't apply as far as the media and the Democratic Party establishment are concerned. Piketty says what they want to hear and that's the only thing that matters.

One way of saving Piketty's conclusion within the above framework would be to permit depreciation to sometimes be negative savings for one class of agents which, by some bizarre convention re. imputed rents, counts as positive savings for another class of agents.
Then what you actually have is two different populations with something like a speciation event occurring- i.e. there is no driver for 'canalisation'.
Perhaps it's the visceral feeling that the Rich really are now a different Species- like in the classic 1992 film 'Society'- which explains Piketty's appeal.
For my part, I thought Growth theory had died an unlamented death in the Sixties.
To be clear, Accounting Identities can always be made to line up if you are Paranoid enough.

Why has this comment not been deleted though this reply to it will be? Surely this is a violation of ergodicity and a clue to why this post is clueless.

Tabaroff's approach only holds if all agents are homogenous so no Schutzian ideal type heterogeneity obtains- more particularly, if some Schutzian Identity classes are heteronomous in the sense of following a rule rather than rational Utility maximisation.
Because Piketty is talking about 'net Income' Tabaroff's approach gains no purchase because we don't know in advance what the future will hold and, therefore, what is or is not 'sustainable'.
More generally, 'Capital', 'Wealth', 'Income', 'Savings' etc are all essentially contested concepts because the future is unknown. Thus, it is purely a Scholastic matter- independent of Empirical evidence- if a particular model sets r (rate of return) as some arbitrary constant such that the Accountancy Convention re. Savings widely diverges from whatever we have grown used to expect.
Put simply, a guy who thinks he isn't saving anything but who lives in his own house might be judged (indeed, is judged, by current Anglo-American National Income Accounting Conventions) to be saving 'imputed rent', and thus getting Wealthier.
There will always be a way to use something like 'Simpson's paradox' to give a Rational Expectations complexion to this.
However, Piketty- I imagine- is in a Malinvauldian, Schutzian tradition which is open to Mimetic effects of a Tardean or even Girardian form. In other words, there is a genuine problem of translation here.

In Mathematics 'mixing'- even weak mixing- is a stronger notion than ergodicity. Samuelson's remark, equating Ergodicity and Econ should be updated bearing this in mind.
Put simply, both Piketty and his critics are irrelevant.

In the steady state, we will all be pretty fucked if g drops to zero. That is some middle ages stuff. Realistically, if population growth is zero and the technological component is 1.5-2.0% at the cutting edge you are not going to get g into the range where this issue becomes a big problem.

@Brad DeLong,

I don't know what the correct depreciation rate is, but here's one quote from p. 43 of Piketty's book.

"This depreciation is substantial, today on the order of 10 percent of GDP in most countries"

Until you figure out the difference between "depreciation rate of 10%" and "depreciation on the order of 10% of GDP" I suggest you stop posting on economics blogs. YER DOING IT WRONG

If you actually look at Piketty's book, you'll see he repeatedly uses 10% as a rough approximation for the depreciation rate, just as Krusell and Smith do. On page 178 of the Kindle edition he writes;

"The difference is important, because annual capital depreciation in the developed economies is on the order of 10– 15 percent of national income and absorbs nearly half of total savings,"

(FWIW, I actually disagree, with Krusell and Smith, for reasons explained in my comment below.)

See, that phrase "nearly half of total savings" is what should clue you into the fact that "the depreciation rate" is AS A PERCENTAGE OF THE CAPITAL STOCK while "10-15 percent of national income" is AS A PERCENTAGE OF NATIONAL INCOME. But hey, what do I know?

Ah, I see. You are correct. I did not catch that important distinction, thanks. I thought you were focused on the word "order of" language.

Marshall, your reply is insightful but unnecessarily harsh.

Apparently not insightful enough! Since it prompted stubborn re-statement of the error rather than self-examination.

Maybe because it was unnecessarily harsh :) .

Oh, but that doesn´t fit his choices, so it´s a worthless fact. Brad is a pragmatic guy, he just sticky to things that support his beliefs, not the other way around, never.

No, actually Brad is correct, since I missed the distinction pointed out by Marshall.

Good you recognize your misunderstood, props to you and Marshall, but you´d better check Alex source before saying Brad is right.

Maybe the world people are afraid is one where the productive capital is all human capital (which gets transferred to the progeny of successful people in one way or another, so has zero depreciation, or maybe even negative depreciation - "learning").

I don't know the correct rate, but depreciation has increased sharply in the US in recent decades because information technology has a much shorter useful life than traditional capital equipment. In the real GDP accounts information technology now accounts for about 50% of business fixed investment versus essentially zero prior to 1980.

Piketty's predictions are indeed based on the assumption that net savings will be constant over the 21st century. But this makes perfect sense when you consider that he *also* predicts that the net rate of return to capital will not fall substantially over the 21st century. If the net rate of return doesn't fall, why should the net savings rate fall?

It's true that to reach a zero growth steady state, you would need to have the rate of return on capital eventually fall, and then presumably net savings rates would fall as well. But Piketty makes no prediction about this because he doesn't expect growth to go to zero, or net return on capital to fall substantially.

It is his *return* assumption that is extreme, not his savings assumption. We're back to the Rognlie critique.

Where in Piketty's text is it made clear that he is providing a model of a "steady state" with an equilibrium condition of k/y = s/g? I can't find that in the book.

I don't think it's in the book. He says that as long as s/g < K/Y, then K/Y will grow, which is correct. He has little to say about exactly how s might eventually change if the rate of return r were to eventually fall. Because he doesn't expect it to fall.

Maybe not in the book, but in this paper it is clear (paper page 13, pdf page 14):

"In the long run, with a fixed saving rate s(t)=s and growth rate g(t) = g ,the steady-state wealth-income ratio is given by the well-known Harrod-Domar-Solow formula: beta(t) => beta = s/g"


Piketty is full of hooey but Alex's first criticism here is a mistake.

Alex writes: "Piketty’s saving assumption is that I=dK + s(Y-dK). What this means is that people look around and they see a bunch of potholes and before consuming or doing anything else they fill the potholes, that’s dK. (If you have driven around the United States recently you may already be questioning Piketty’s assumption.) After the potholes have been filled people save in addition a constant proportion of the remaining output, s(Y-dk),"

No, don't be silly. Piketty is merely saying savings to mean net savings, whereas the tradition among economists to say savings to mean gross savings. That's very important to keep in mind but shouldn't be all that troubling. It does not in any way imply that Piketty thinks people are under a compunction to repair all wear and tear before investing. Net savings obviously always includes some wear and tear of things that aren't repaired during the period and likely never will be. Piketty knows that. He's a shambolic theorist but he's not stupid.

It is more than "merely saying savings to mean net savings" if you assume net savings to be constant.

That's already an element of the formulas that turn out to be silly. Actually that's his savings rate defined as net savings to net income. Modeling that rate as a constant is obviously very far from realistic and bound to mislead. But doing so does not at all imply any prioritizing of maintenance over new investment. Alex made that up out of thin air and he should correct it.

Its a folksy way to say it, but its not very misleading. No matter how high depreciation is, you cover it with savings in addition to the new net investment given a fixed net savings. In other words, gross savings must perfectly offset an increase in depreciation to keep the net the same. After that you consume whats left. Sure the order of the replacement of worn out capital and new capital isn't specified in the model, but he (Piketty) is apparently assuming that all depreciated capital is replaced every period, so that there is always a fixed ratio of new net investment. The model prioritizes maintenance over consumption. That is more than specifying everything in net terms.

Tabarrok and @youngecon are correct. Piketty assumes a constant net savings rate out of net income s=(I-dk)/(Y-dk), reworking that you get Tabarrok's statement..

It's unfortunate Alex made that mistake, because otherwise the piece is good. Piketty is waving around seemingly sensible formulas that when analyzed turn out to be silly. The growth model that has capital/output escalating with declining growth is one such case. It's neither substantiated by any empirical data nor logical from a rational maximization of utility point of view.

Not substantiated by any empirical data? What about ALL THE PRE-20TH-CENTURY DATA PIKETTY REPORTS IN THE BOOK? (Not to mention The Return of Capital). The main argument for Piketty's prediction about the rise of K/L isn't that some representative agent theory says that's what happens when g-->0; it's that that is what indeed happened when g was closer to zero (ie, before the 20th century). Krusell and Smith (and Tabarrok) are stubbornly waving around their macro theory received wisdom in the face of Piketty's evidence, to which they have no reply now that the FT ploy didn't quite work out.

I repeat: there is no evidence that what Piketty calls the capital/output ratio, essentially private net worth to output, consistently escalates when growth slows. Sometimes it grows, sometimes it doesn't change much, sometimes it shrinks. There are lots of post-war examples from emerging markets where the private net worth to output ratio rose as growth quickened.

If your argument is that some 19th century or earlier "data" shows that the private net worth to output ratio did escalate during some historical slow growth episodes, first you're going to have to convince people that your so-called data is real and reliable and shows a consistent behavior in many different times and places. Given that output and net worth data that old must be largely guessed, that task seems a bit Herculean. And if you succeed at it, you're going to have to produce an argument why private net worth to output rose when growth fell pre-20th century, showed no consistent pattern since, but is bound to revert to pre-20th century behavior.

* should have said the rise of K/Y, not K/L in the above.

Here's how I interpret Piketty's inherently dynamic model of wealth and income growth.

Piketty's savings rate for any year i is [W(i+1) - W(i))]/Y(i). It is simply the change in wealth as a percentage of national income. Wealth and income are the more fundamental concepts.

Y in the above is net national income, which he defines as gross output minus depreciation, plus net income from abroad. Write it as follows:

Y = (GDP - D + A)

Then we get W(i+1) = W(i) + s*(GDP(i) - D(i) + A(i))

Wealth is a very inclusive concept for Piketty. It includes the stock of fixed capital goods, but it also includes (among other things) any surviving inventories of consumption goods that have not been consumed yet. Let's leave out the "other things", and use "KW" and "CW" to refer to fixed capital wealth and consumption good wealth respectively, and decompose W into those two components. Then we get:

KW(i+1) + CW(i+1) = KW(i)+ CW(i) + s*(GDP(i) - D(i) + A(i))

Suppose we identify net investment with the net change in fixed capital wealth, so I(i) = KW(i+1) - KW(i). Then we have:

I(i) = CW(i) - CW(i+1) + s*(GDP(i) - D(i) + A(i))

Now I see no way to derive any conclusions from this about the rate of investment relative to the rate of depreciation (no matter which concept of depreciation is used). A society can build its wealth from one year to the next by producing and hoarding consumption goods, even if it suffers a net loss of fixed capital stock. It can also build wealth by receiving positive net income from abroad and saving some of it. If a society is a wealthy net rentier, this aspect of wealth-building could be substantial.

(Any inferences about investment are made even more iffy by the fact that there is no hard and fast distinction between capital goods and consumption goods. Also, in leaving out the "other things", I left out circulating factor inputs which are not fixed capital, but are also not consumables.)

What about the long run? I don't think Piketty offers an equilibrium model of growth in the manner of Solow. The one section of the book that deals with such matters, Chapter 6, I interpret to be ad hominem: that is, Piketty is just trying to show that even in the customary framework, an economy can experience a volatile and growing capital share with relatively small elasiticities greater than 1- and doesn't require elasticities going to infinity. His conclusion to the chapter is that there is no mechanism inherent in capitalism that guarantees either a reduced or stable capital share as wealth accumulates relative the national income.

My suggestion is that people throw out most of what they think they know about Piketty based on their adaptation of pre-existing models, and build up Piketty's model from scratch.

Is g government spending? Growth?

Just as an aside, it's worth remembering that Keynes did something sort of similar in his version of the multiplier as explained in the General Theory. He proposed that the propensity to consume, a kind of inverse of the savings rate, was essentially constant. Therefore, he theorized, public investment at less than full employment would spur additional consumption in proportion to the propensity consume, because the propensity to consume was assumed to remain constant. This is usually called the Keynes-Kahn multiplier, since Keynes adapted it from Richard Kahn. It got dropped quickly in Keynesian theory which came up with other logic for multipliers.

And it wasn't any more correct when he did it than when Piketty does it. Wrong ideas are wrong no matter how old they are.

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