### Chamley-Judd revisited

9/17/2014 04:52:00 PM
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When I started grad school we started with a math boot camp, of which the first topic was formal reasoning. One of the exercises that some of my classmates struggled with was counter-factual reasoning. Take for example the theorem:
If 1+1=3, then 2=1.
This is a valid theorem. We can prove it: we know that 2=3-1, and it is postulated in the theorem that 3=1+1, therefore 2=1+1-1, which implies 2=1. Neither the postulate nor the conclusion are factually correct, but the theorem is nevertheless correct.

I mention this example because it turns out that one of the most important theorems in tax theory makes exactly such an error. I'm referring of course to the famous Chamley-Judd result which is usually described as saying that we can't redistribute capital income to workers--that the optimal tax rate on capital is zero.

But that's not what the theorem actually says. Take for example Judd (1985): in the rather extreme setting where capitalists are unable to work and workers are unable to save, Judd's theorem says
Theorem 2. If the redistributive capital taxation program maximizing a Paretian social welfare function converges [to a steady state]...then the optimal capital income tax vanishes asymptotically. Specifically, there should be no redistribution in the limit and any government consumption should be financed by lump-sum taxation of workers.
Ok, there's a lot of verbiage in there, but the point is that Judd's theorem postulates that at the optimal tax rate all quantities and multipliers converge to fixed steady state values, and from this concludes that that optimal capital tax rate is also constant at zero. As a theorem, "if steady-state exists, then optimal rate is zero" is correct. But a recent paper by Ludwig Straub and Ivan Werning showed that Judd's postulate "if a steady-state exists" turns out to be about as wrong as 1+1=3. As a result, so is Judd's conclusion.

In fact, at the optimal tax rate the steady-state which Judd assumed does not necessarily exist. To illustrate the conditions when it doesn't exist, suppose that capitalists have utility functions of the form $$U=\sum_{t=0}^\infty\beta^t\frac{C_t^{1-\sigma}}{1-\sigma}$$ (which is the same as in my DSGE calculator) where $C_t$ is the capitalist's consumption in period $t$, and $\beta,\sigma$ are just constants. As Straub and Werning showed, if $\sigma\gt 1$ and we want to make workers as well of as absolutely possible, then the optimal tax rate doesn't converge to zero as Judd claimed, but actually diverges all the way to 100 percent1 in the long run! It turns out that at the actual optimal tax rate, equilibrium doesn't converge to a steady-state but actually diverges so that capital stocks and consumption plummet towards zero over time. This result illustrates exactly how extreme a setting the Judd model where workers can't save really is--so extreme, that workers are actually better off suffering immiseration than the pittance they'd earn under a zero-tax regime. At least with immiseration, workers will get to consume the capital stock first.

Judd is still wrong in the special case where $\sigma=1$ which corresponds to logarithmic preferences.2 Even though in this case the quantities do converge to steady state, the optimal capital tax is still positive in the steady-state because it turns out that the multipliers diverge. If you're unfamiliar with the math, 'multipliers' are a weird relic of mathematical optimization techniques that do not represent real-world things--they can be interpreted as the theoretical "marginal utility of wealth" but they are really just abstract mathematical constructs. Yet Judd's theorem requires that these too converge to steady state values at the optimum tax rate, which isn't even true in the simplest case of his model.

For anti-taxers, there is a small silver lining. If $\sigma\lt 1,$ then the equilibrium does converge to steady state and the optimal tax rate converges eventually to zero. But for practical purposes, even if you think $\sigma\lt 1,$ this probably doesn't matter much, because convergence to that steady state is quite slow:So, no, we can't actually say that the optimal tax rate on capital is zero. Chamley-Judd didn't even say that!

I've only focused on Judd, but Straub and Werning also look at Chamley as well. There's a lot more in their paper than I was able to squeeze in here, so go check it out!

1 I'm assuming that the government's only function is to redistribute. In the case where the government also consumes resources (often termed "wasteful government spending"), such as when redistribution imposes administrative costs or where government provides services other than social insurance, then the long-run capital stock must remain just large enough for the government to be able to finance it's own consumption, which requires a maximum long-run capital tax less than 100 percent.

2 To see why, take the derivative: if $\sigma=1$ then $$\frac{\partial}{\partial C_t}\left(\frac{C_t^{1-\sigma}}{1-\sigma}\right)=\frac{1}{C_t}=\frac{\partial}{\partial C_t} ln\left(C_t\right)$$ for all $C_t.$
Luis Enrique 9/18/2014 04:47:00 AM
thanks for this very clear exposition