Alex Tabarrok interprets Thomas Piketty’s model of wealth and income dynamics in terms of a “super-simple” Solow growth model with the steady state requirement that investment is equal to depreciation. And he concludes on that basis that Piketty is committed to the idea that the economies he is modeling always invest in sufficient amounts to offset any depreciation, prior to any additional consumption and any additional saving. This is what he says.

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.

As I have indicated previously, my strong suspicion is that this is a quite wrong-headed approach to understanding what Piketty is up to, and that any attempt to read Piketty as putting forward some kind of steady state equilibrium growth model distorts his argument.

Here’s how I interpret the rudiments of Piketty’s inherently dynamic model of wealth and income growth:

Piketty’s national savings rate s_i 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 Piketty defines as gross output minus depreciation plus net income from abroad. Write it as follows:

Y = (GDP – D + A)

So 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” for now, 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 net 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. On the other hand, a society can lose wealth, even if it net invests in excess of depreciation, if it decides to consume a large enough portion of its stock of consumable wealth, or if it is a net rent-payer to its landlords abroad.

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 used by most macroeconomists, an economy can experience a volatile and growing capital share with relatively small elasticities slightly 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 the pre-existing models they are carrying around in their heads, and build up Piketty’s model from scratch. The ingredients are all there in the text.