Spreadsheeting r>g

01/05/2014 § 3 Comments

Thomas Piketty’s book, Capital in the 21st Century, has been getting press, some favourable, some lukewarm, some critical. There’s even a bluffer’s guide (bonjour paresse!).

I haven’t read it, but that won’t stop me commenting. Specifically, the little shorthand ‘r>g’ making the rounds had me thinking. Unfortunately, my thought is also the first entry in the bluffer’s guide: the thesis isn’t new. This is the tendency for the rate of profit to fall, by some 19th century economist, dressed up a different way. It’s also something my actuary/economist dad pointed out to me years ago — that stock market returns couldn’t keep outpacing economic growth forever. And something that can’t go on forever, won’t.

But it isn’t real until you can put it in a spreadsheet. So, I tried. My first attempt failed because ‘r>g’ isn’t enough by itself.

So, I tried again, this time including a marginal propensity to save, which you need in order to determine how much income gets converted into wealth. It turns out to make for interesting calculations.

Here’s one example. Start with GDP = 100, divided 60/40 into wages and rents. Assume g = 0.02 and r = 0.08. With the amount and rate of rent, you can calculate initial capital (K), which is here 40/0.08 = 500.

GDP Wealth
g= r= MPS_L= MPS_K=
0.02 0.08 0.1 0.8
Year total wages rent savings, L savings, K Final K
1 100 60 40 6 32 532
2 102 59 43 6 34 566
3 104 59 45 6 36 602
4 106 58 48 6 39 641
5 108 57 51 6 41 682

What happens in the second period depends on what happens to rents. If they are entirely consumed by dissolute third-generation scions, then they don’t add to the stock of capital. So period 2 depends on the marginal propensity to save, which here I’ve assumed is 0.8 (80%). Final K is higher than the initial K, and the amount of rents increases. The result over many periods is the following:


The picture, though, is sensitive to the assumptions. Assume g = 0.03, r = 0.08, and MPS_K = 0.4, and here is the 100-year picture:


It turns out that the results depend on initial allocations, relative rates of returns, and savings rates. Crucially, too, I haven’t actually created a stock of K wealth that is owned by the initial L. I created the category in the spreadsheet, but then didn’t use it. If labour starts owning bits of capital, well, either that’s employee-owned companies or control of the means of production by the proletariat — I’ll let you make the call.

It seems that the problem is prying rents out of the hands of capital-owners, rather than the rate of rent itself. One way, of course, is taxes. A 50% estate tax looks like a useful way to get MPS_K from 0.8 to 0.4, for example. And dissolute grandchildren should also be encouraged.

Happy May Day!


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§ 3 Responses to Spreadsheeting r>g

  • Donal Curtin says:

    Hi Bill – if your comment that “something that can’t go on forever, won’t” is your own, good on you, or it could be an erudite reference to Stein’s Law, but if it’s not, Stein’s Law says “”If something cannot go on forever, it will stop”

    • Thanks, Donal. Yes, it was a reference to Stein’s Law, but I didn’t look up the actual words. So, more a lazy reference than an erudite one! But it’s important to remember that something unsustainable won’t be sustained.

  • Matt Nolan says:

    Nice post. A key underlying assumption in all of this is how the combination of capital and labour creates output – the more complementary the factors are, the higher w will end up being.

    Long run g is only a constant as the accumulation of capital, and movement of labour, has adjusted such that the relative rates of return have been equalized – and we only get this out by having a relationship between these factors. To me this poses the most interesting bit of the entire conversation!

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