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.