### Waldman on Capital Gains Taxes

3/21/2013 03:04:00 PM
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Steve Randy Waldman has an excellent post on capital taxation. You should read it. Really.

At issue is a theoretical result by Christophe Chamley in 1986 and Kenneth Judd in 1985. These two papers looked specifically at the question of whether we can get redistribution of wealth through taxation. In the extreme case, they supposed that there were two types of agents: capitalists own do all the investment but don't work at all, while laborers supply all the labor but don't invest at all. Even in that world, Chamely and Judd found that it was always optimal to eventually reduce the capital tax rate to zero. Intuitively, since capital and labor are complements (and production is constant-returns to scale), taxing capital reduces the capital stock, which in turn reduces worker's wages.

Here's my take on the issue, elaborating and adding to Waldman's objections to Garret Jones's post. First, I clarify that a couple of Waldman's critiques are actually a statement of the fact that when there are nontaxable inputs in production, capital should be taxed. Second, I discuss the fact that Garret Jones actually had to ignore part of the Chamley-Judd results to assert that capital taxes should be zero, since it is optimal to tax capital when we are away from the steady state. In the third and final comment, I point out that Chamley-Judd only applies to flat-rate taxes, and that the research optimal nonlinear taxation show that it is often optimal to tax capital at highly progressive rates.

## Ignores the Problem of Incomplete Taxation

Waldman points out that this result depends on the assumption that production is constant returns to scale:
...the optimality of a zero capital tax rate is in fact a “knife-edge” result with respect to returns-to-scale of the production function. If returns to scale are decreasing or increasing, the optimal capital tax rate from workers’ perspective may be positive or negative, and is sensitive to details. A stable, constant-returns-to-scale production function may be attractive for reasons of convention and convenience, but it is unlikely to be true. Once we admit this, we really don’t know what the optimal tax rate is in an otherwise Chamley-Judd world.
This is a slightly different way of stating a standard text book result (page 495 of Sargent and Ljunqvist, second edition). Now, what Waldman means here is the returns to scale specifically with respect to the inputs that the government is capable of observing and taxing--there could be unobserved inputs that makes the overall production function linearly homogeneous. If there is decreasing returns to scale with respect to capital and labor, then the optimal capital tax is greater than zero. Waldman goes on to say that
As Jones hints in his “bonus implication”, labor is not in fact measurable in terms of homogenous hours....Macroeconomically, our collective capacity to produce improves. You might, as Jones does, refer to this incorporeal je ne sais quoi that enhances labor over time as “human capital”, or as labor-augmenting technology
I guess Waldman was unaware, but whether or not "human capital" accumulates or is bounded or whatever is completely irrelevant. And the argument can be generalized to any input, including black-market labor or variable workers' effort, not just human capital. We only need to establish two facts: 1) the input and physical capital are complements in production and 2) the input cannot be directly taxed. This ties into Waldman's earlier point since if there is an additional input, then a constant returns to scale production function overall is decreasing returns to scale in capital and labor specifically. The result is that when there is a nontaxable input in production that is complementary to capital, then it is optimal in the long-run steady state to levy a strictly positive capital tax rate. At the very least, you have to admit that in our service-dominated economy workers' effort is variable and cannot be directly taxed. That fact alone means we should tax capital, even in a Chamley-Judd world.

## Considers Only the Non-Stochastic Steady State Tax Rate

I have an additional criticism of using Chamley and Judd to argue for a zero capital gains tax, even when we assume the world actually resembles the Ramsey model: Chamley and Judd only showed that the zero tax rate is optimal in the non-stochastic steady state. When we are not in the steady state, according to Chamely,
"The previous result shows that there are two regimes for the interest tax. The policy is either to tax as much as possible or not at all. The tax has two effects. [In the first regime when $t\geq\tau$] It raises revenues on existing capital but it also introduces intertemporal distortions in saving. The lump sum effect of the capital tax overrides the savings distortions for relatively small values of $t$ ($t < \tau$). In the second regime, the saving distortions become the predominant factor...there is no interest tax...and the government generates revenues only by taxing wage income (or other commodities in a multi-good economy)."
What this says is that in the Ramsey model it is actually optimal to tax capital at 100% for a "short" time period when the economy is away from the steady state. Chamely does a back-of-the-envelope calculation and finds that "short" could reasonably be in the range of 6 to 8.8 years, or longer. The intuition for the result is this: in the short run, increasing the capital gains tax will tax existing capital more than new capital formation. By taxing existing capital and lending the proceeds to the public (paying down the debt and/or acquiring private-sector assets), the government can decrease the need for future revenues. The optimal tax plan actually involves trying to tax existing capital as much as possible before eventually reducing the tax rate to zero. Garret Jones seems to have missed that. Now, in reality the economy is stochastic, and rarely in the steady state, so it turns out that the optimal capital gains tax rate is variable (could be positive or negative, depending), and rarely zero.

## Ignores the Possibility of Nonlinear Taxation

This second criticism actually hints at the larger problem with the Chamley-Judd model. (at least, it does to me, you might see this as a non-sequiter) Namely, by relying on the Ramsey framework, Chamle-Judd are only able to look at linear flat-tax rates on capital. That is a terrible, terrible way of looking at tax policy in models with heterogeneous agents. In reality, we can charge different agents different tax rates by, for example, having progressive, non-linear tax rates. Hence, it is possible to levy a zero capital tax on, say, poor people while levying a large positive capital tax rate on the rich. When you adjust the framework to allow non-linear taxation, this turns out to be optimal. I have proposed a greatly simplified version of this non-linear tax plan in a post here.

By ruling out the possibility of non-linear taxation, Chamley-Judd actually tell us very little--they merely told us that out of infinitely many possible tax policies, a few of them (the linear non-zero ones) are sub-optimal in the special case of a non-stochastic steady state with complete taxation and constant returns to scale.