Be careful how you wield Chamley-Judd

4/29/2014 03:29:00 PM
Don't bring a gun to an economics theory seminar.
Yesterday I illustrated the "immiseration" principle that suggests that, absent intervention, inequality will increase without bound and as a result, almost everyone will become impoverished while a single dynasty takes all the wealth. The result is surprisingly robust in theory. Today I find Forbes arguing that, even if Piketty is right to be worried about immiseration, there's simply nothing we can do about it--the famous Chamley-Judd result proves that there is nothing we can do.

This is all kinds of wrong, and is a good opportunity to illustrate that while "immiseration" is a fairly robust result, Chamley-Judd is not. For reference, Chamley-Judd is a theorem dating back to the 1980s saying that in a Ramsey setting in steady-state with linearly-homogeneous production in which capital and labor are complements, where we have complete taxation of all factors of production, and a couple other things, then the optimal linear tax rate on capital eventually goes to zero. In fact, it says that in the long run, any linear tax on capital will reduce workers' wages by more than the revenue raised by the tax. Of course, any graduate macro text will show you some of the ways in which Chamley-Judd assumptions are violated in reality, producing a non-zero optimal tax rate. Here's one example. Steve Randy Waldman listed some other ways here and here, with readers in the comments contributing a long laundry list of ways in which Chamley-Judd has been overturned over the years. But this post isn't about that (I rather cryptically foreshadowed today's post back then--it's day has finally come). Let's play by the rules and suppose that Chamley-Judd holds in it's original form.

Actually, Forbes has gravely misinterpreted Chamley-Judd. Basically, Chamley-Judd says it's hard to redistribute from capitalists to laborers. But it's actually exceedingly easy, without violating Chamley-Judd, to redistribute from rich to poor capitalists.

To illustrate, Let's return to the theoretical setting I laid out in yesterday's post. What happens in that model is that one dynasty's share of the total wealth increases without bound, eventually owning all of it, while all other individuals eventually starve to death. Side-stepping Chamley-Judd, while also preventing the immiseration result, is as easy as this: just impose a flat tax on capital gains with a high standard deduction. Individuals with capital gains below the deduction amount still face a zero marginal tax rate, meaning that with this non-linear tax, the poor investors have a permanent advantage compared to rich investors. As a result, over the long run, rich dynasties will decrease their capital stocks so that they are at the deduction level, while poor dynasties will gradually increase their capital stocks. In the long-run, the tax doesn't actually generate any revenue, but nevertheless makes the poor way better off and prevents immiseration. Because at the margin in the long run the marginal tax rate is still zero, Chamley-Judd is not violated at all.

In fact, Emmanuel Saez has a paper that says exactly this, and Forbes would be wise to check out something published at least in the past two decades before criticizing Piketty.

Chamley and Judd did two things differently than our example above: first, they assumed that there was a class of individuals (workers) who were totally unable to invest in capital. Second, they assumed that any tax on capital had to be linear--that is, without any kind of standard deduction or progressive graduation. Removing either, or both, of these restrictions allows us to use the tax code to redistribute from rich capitalists to poor workers.

Above we supposed that everyone is equally competent at investing. However, let's maintain the Chamley-Judd assumption that there's a class of workers who, for whatever reason, are totally unable to invest in capital. But we will allow for the capital gains (and interest income) tax to be progressive, so that those with little interest income face a zero marginal tax rate. For the rich with lots of investments, the progressive cap gains tax represents a distortion of the consumption-savings trade-off, leading them to invest less of their own funds into capital, resulting in a lower capital stock which harms workers by depressing wages (because capital and labor are complements in production). But, there are still tons of workers in this economy who face a zero tax rate on interest income. This opens up an arbitrage opportunity for the rich: they can borrow from the workers and invest the borrowed money into investments, and pay their higher capital gains tax only on the net returns after interest is paid out. The high tax on large capital gains no longer matters to these rich capitalists, because the income from arbitrage is free money; borrowed funds do not subtract from their consumption, so the tax does not distort their consumption-investment trade-off. This progressive capital gains tax benefits poor workers in two ways: first, it shifts the risk of investment off the poor worker and onto the rich capitalists--risk, as you'll recall from yesterday's post, represents a bigger problem for the poor than for the rich, so this represents a net increase in aggregate welfare. Second, this progressive taxation lets workers share a substantial portion of the returns from capital without a commensurate loss in wages--we do, in fact, see a net redistribution from capitalists to workers. Note that, as before, the point of this progressive cap gains tax is not necessarily to generate revenue to be distributed, but rather to cause redistribution of pre-tax income in the marketplace.

Bottom line is this: Chamley-Judd ain't got nothing on Piketty.