# Wealth and Capital are Different Things

Piketty’s book is like a giant attention-sucking vortex. I can’t seem to escape it. This time I’m thinking about the criticism of Piketty’s analysis that has to do with rates of return on capital. Piketty says that if ${r > g}$, where ${r}$ is the return to capital, and ${g}$ is the growth rate of aggregate GDP, then wealth will become more and more concentrated.

Critiques of Piketty have questioned the assumptions underlying this conclusion. The most recent one I’ve seen is in Larry Summers’ review piece. Let’s let him sum up the issues:

This rather fatalistic and certainly dismal view of capitalism can be challenged on two levels. It presumes, first, that the return to capital diminishes slowly, if at all, as wealth is accumulated and, second, that the returns to wealth are all reinvested. Whatever may have been the case historically, neither of these premises is likely correct as a guide to thinking about the American economy today.

With respect to the first assumption regarding the rate of return, here is what Summers says:

Economists universally believe in the law of diminishing returns. As capital accumulates, the incremental return on an additional unit of capital declines.

But Summers has fallen into what I think is a really common trap for economists. He presumes that his second statement (“As capital accumulates, the incremental return on an additional unit of capital declines”) contradicts Piketty’s assumption (“that the return to capital diminishes slowly, if at all, as wealth is accumulated”). These two statements are not mutually exlusive.

The issue is that Summers is confounding wealth and capital. This is not helped by Piketty, who uses “capital” in his title and in the book the way that normal people use it, as a synonym for “wealth”. But from the perspective of an economist, these two concepts are not the same thing. The capital that Summers refers to in his critique (often denoted ${K}$) is a subset of the measure of national wealth (${W}$, as I’ll call it) that Piketty documents.

Without going too deep into this, Piketty’s measure of wealth consists of three parts: real estate, corporate capital, and financial assets. Only real estate and corporate capital are what economist have in mind when they say capital (${K}$). Wealth, however, consists of all three parts, so that Piketty’s wealth is ${W = K + F}$, where ${F}$ is the value of financial assets. Asserting that the return to capital falls as the capital stock increases – as Summers does – does not imply that the return to wealth falls as the stock of wealth increases. Even if we assume that financial markets work so efficiently that the return to capital and the return to financial assets are identical, this does not mean that the return to wealth necessarily falls as wealth accumulates.

To see this, consider a really slimmed down version of the “bubble asset” model from Blanchard and Fischer (1989, p. 228). We have that the return on capital is ${r = f'(K)}$, where ${f'(K)}$ is the marginal product of capital. The ${f'(K)}$ is the derivative of the production function, and represents the marginal increase in output we’d get from adding one more unit of capital. Under our typical assumptions about diminishing returns, as ${K}$ goes up ${r}$ goes down. This is what Summers is using as his critique.

An efficient financial market would ensure that financial assets (F) would also have a return of ${r}$. If they did not, then people would buy/sell financial assets until the return was equal. (Yes, I’m ignoring risk entirely, but that doesn’t change the main point here). So the return on all wealth is equal to ${r}$, and note that this is pinned down by the value of ${K}$ alone.

Now, we have assumed that ${r}$ falls as ${K}$ increases. Does this imply that ${r}$ falls as wealth (${W}$) increases? No. The relationship between ${r}$ and ${W}$ depends entirely on the composition of the change in ${W}$. If ${W}$ rises because ${K}$ rises (say ${F}$ stays constant), then the rate of return on wealth falls because the marginal product of capital has declined. This is what Summers and others have in mind.

However, it’s perfectly plausible that ${W}$ rises even though ${K}$ falls, because the value of financial assets (${F}$) are increasing even more quickly. In this case, the marginal product of capital has increased, and the rate of return on wealth has increased. In this case, the rate of return rises with wealth.

Is it reasonable for an economy to experience falling capital but a rising value of financial assets? Sure. The point of Blanchard and Fisher’s model of bubbles is that even though all individuals are acting rationally at all times, the economy can take off onto a weird path where the stock of capital (${K}$) gets run down while the value of financial assets (${F}$) rises. Eventually this is unsustainable, as we’d run out of capital, but there is no reason that a situation like this cannot persist for a while.

Will the return to wealth necessarily rise as wealth accumulates? No. There are other equally reasonable paths that the economy could take where wealth accumulation is driven mainly by capital accumulation and the rate of return falls as wealth accumulates, consistent with the Summers critique. The point I want to make is that there is no particular reason to believe in a fixed relationship between wealth and the return on capital. They can move completely independently of each other.

So Piketty can easily be right that we are currently in a world where both the wealth/income ratio is increasing and the rate of return on wealth is rising (or remaining roughly constant), and that this could persist for some indefinite period. On the other hand, it was not inevitable that this was going to happen, and it could just as easily end tomorrow as in 100 years.

I think the story that is milling around beneath the surface of Piketty’s book is that recent wealth accumulation has been primarily of financial assets, not capital. Hence the return has stayed high and the concentration of wealth has continued. If the returns on that wealth are continually reinvested in financial assets as opposed to capital, then Piketty’s death spiral of wealth concentration would likely be the outcome. To avoid that death spiral, you’d want to get the returns on wealth reinvested into real capital so that the return on capital (and hence wealth) gets pushed down.

## 10 thoughts on “Wealth and Capital are Different Things”

1. This is an interesting and important point, and you have made it very clearly. However, it seems to imply that the way Piketty’s hypothesis can survive Summers’ critique is through a financial asset bubble that is inherently unsustainable. But Piketty’s claim is to a long-term and inherent tendency of capitalism, not some transient rational bubble phenomenon. The conclusion still seems to be that the proposition “r > g” cannot persist indefinitely and that there are natural forces that work against it over the longer run (through some combination of the popping of a bubble and diminishing returns to capital)?

Regardless of the nitpicking, this raises an interesting policy prescription: if inequality and the persistence of “r > g” are dependent on financial asset bubbles, then tackling bubbles would help mitigate inequality. Of course, tackling bubbles is easier said than done (as is tackling inequality by most other means)…

• I’d have to think about it a little more, but I think you’re right that the growth in the financial asset would ultimately be unsustainable, and some kind of crash would be unavoidable. But I think the “bubble” in the financial asset in my little toy model is different than what we typically think of as “bubbles”. What I mean is that you could get growth in financial asset values in general without being able to look at any specific asset class (real estate, tech stocks) and have it look wildly over-valued.

It’s also not necessarily true that financial asset values would rise and rise forever until they popped. We could be transitioning from a mid-20th century steady state with high K and low F to a new steady state with low K and high F. But once we hit that new steady state the process will settle down and F will stop growing. That’s probably the way I would make the strongest pro-Piketty argument using my little theory. In the 80’s/90’s something happened (financial deregulation, etc..) that has fundamentally changed the steady state for the economy, and we are headed towards that new steady state where wealth is (a) very large relative to GDP and (b) is composed of a larger fraction of financial assets compared to physical capital. So I don’t think Piketty *needs* there to be a bubble to be right.

That being said, maybe what we see in Piketty’s data is actually a bubble, and there will be some kind of bubble popping event like WWI or WWII that resets wealth/income ratios to a lower value in the future.

• I was about to leave a comment about the whole long run-short run distinction as well, but was clearly beaten to it. I’ll just add an additional point/question, however: as you state yourself, “[a]n efficient financial market would ensure that financial assets (F) would also have a return of {r}.” In the long run, I think we can agree that we expect financial markets to equilibrate, so that r_{fm} = r_k. But that’s all you need: if it is true that r_k = f'(k) and f”(k) < 0, then dr^{ss}/dk < 0 (where I've used the conventional steady state, SS, to denote the long run), which should also be consistent with dr^{ss}/dW < 0, no? I don't see how F and K and move in opposite directions indefinitely in the steady state.

• K and F are different types of assets, so they can move independently over some time frame, if not indefinitely. In the Blanchard and Fischer bubble model, F can be growing while K is shrinking along some transition paths. Those paths explode to infinite F and zero K if you follow them out to their logical conclusion as time goes to infinity. But you could be on that path for a very long time before it becomes obvious it is unsustainable. There is nothing that says the world has to be on a transition path towards to a stable steady state. It’s just something we assume.

2. Not often mentioned is that r > g can also be created by a steady drop in g, or at least the fraction of g permitted to go to the general population.

Noni

• I think you have to be careful here. Piketty’s thesis is precisely that g is falling as population growth slows down, while r is staying roughly constant. By “fraction of g” I think you mean “share of national income”, and Piketty documents that this is rising slightly over the last 30 years. The combination of those two are what drives his conclusion of higher inequality in the future.

• Not so fast. g=fertility rate. The economy expands with the population. Conversely, low fertility equals low g.

OK. what else does low fertility imply? Less heirs. So wealth is distributed to less people implying a concentration of wealth.

So we have some correlations here. Low g certainly correlates with r>g and it correlates with wealth concentration. The trick is to assign causation accurately. r>g correlates with wealth concentration but it doesn’t cause it. low fertility caused everything. I think this is what Summers was trying to explain.

• Not sure what “not so fast” is meant for. In Piketty’s work, g = pop growth + per capita output growth. And yes, as population growth rates fall, so will g.

The question that Summers and others have is why r does not fall along with g? In a model with just productive capital, then r should decline as more capital is accumulated, and the economy should not be able to sustain r>g. It’s naturally equilibrating.

My post was about the fact that the “r” that matters is the return on wealth, and that could well stay bigger than g even as productive capital accumulates.

3. Real estate is not capital in the sense of a factor of production either, since it connsists for the most part of land (a non-accumulable and non-produced factor of production, as Piketty himself acknowledges, and residential housing, a consumer durable). Moreover, to apply the Harrod-Domar condition, capital has to be domestic and only measured in current-producer-price units, not market valuations.

To see just how large this difference is historically, see my blog post http://silverberg-on-meltdown-economics.blogspot.de/2014/06/nitpicking-piketty-productively-part-i.html.

According to the standard growth-accounting way of measuring capital as a production factor (the perpetual inventory method), the capital-output ratio in the US has actually been gradually declining in the US since 1950. There is no decreasing returns to capital problem to address at all, even if you believe in an aggregate neoclassical production function. Moreover, the national capital-output ratio (which excludes domestic debt) is entirely orthogonal to the distribution of wealth among private households, where debt (government and corporate bonds, consumer debt and mortgages) plays a substantial role.

• For the purposes of estimating productivity when doing growth accounting, I may not want to include real estate (except that GDP does include the imputed flow of services from housing, so I actually do). But for Piketty’s purpose – talking about the distribution of wealth and income – I absolutely do want to include real estate as part of wealth. He’s not trying to stick his measure of wealth into some production function, he just wants to establish the ratio of wealth to income. Real estate is wealth.