## Friday, May 30, 2014

### r and g redux

Many people are jumping all over Piketty's claim that r>g  implies growing inequality. I'm going to leave that alone, but I do want to make some points that may or may not be relevant.

Many years ago, Joan Robinson suggested that workers are exploited to the extent that capitalists consume more than zero! Here's why: she 's assuming workers save nothing, so the savings rate will be   arK/Y. where a is the rate at which capitalists save. Since the long-run value for K/Y is the net savings rate divided by g, the growth rate -this is one of Piketty's fundamental laws and is trivial to establish- we then have, in the long run K/Y =(arK/Y)/g, or r =g/a. So when capitalists live on air, a=1 and r=g. If capitalists consume anything, a falls short of 1 and  r>g.

And here's the key:  r=g  is  needed to satisfy the Golden Rule: steady-state consumption is maximized for the representative generation only when r=g.

If you are familiar with the overlapping generations model - the canonical articles, especially Gale (1973)- you know that this is right.  Many people note that r  less than g is inefficient.  But most characterize r greater than or equal to g as efficient. And it's true, if r strictly exceeds g, we have efficiency.  But the reason for the efficiency is that moving towards the golden rule would require harming the initial old  generation. Every other generation, however, will be better off both transitionally and in the golden rule steady state. And what's more - see Gale and also Diamond (1965) - the harm to the initial old can be made arbitrarily small.

I think people don't make this point because they've bought the infinitely-lived agent nonsense, or the Barrovian dynastic equivalent, where the optimal r is the rate of time preference. But as Gale made a point of noting, in an over-lapping generations frame-work, time preference is irrelevant! We need r=g for optimality. Robinson, as usual, was right!

By the way, if you haven't read Fredric deBoer's review of Megan McCardle's review of Piketty - the one which opens with McCardle admitting she hasn't read Piketty- here is a taste which bears on (sort of!)  r and g :

"Finally, there is McArdle’s dubious takedown of Snicket’s statistics. By now, Snicket’s controversial equation “r>g,” where r = points per game from center Nicklas Backstrom and g = penalty minutes/100 for winger Tom Wilson, has become something of a mantra in left-wing circles. McArdle responds with her own equation, which seems much less convincing– s = (bs – epst)*hyk, where s stands for success, bs for number of bootstraps, epst exposure to public school teachers, and hyk the Coefficient of Hayek. While I appreciate McArdle’s focus on hard work, this seems to do little to address the book’s real thesis."

I'm damned if can put the link in, but look around and you'll find the whole thing!