# Optimal Taxation of Capital Income 2019 (let them Bern).

I wrote a post about optimal taxation of capital income which (the web is sometimes wonderful) was made legible by the blessed [person who choses to remain anonymous].

But that was back in Obama center left 2008. I want to update given what I learned since then and given the appearance of socialist US citizens.

First, what I should have known already is that the standard Judd 85/86 result that the optimal rate of taxation of capital income goes to zero as time goes to infinity is what mathematicians call a boo boo (oopsie). The asserted theorem is false as explained by Ludwig Straub and Ivan Werning.

This is an interesting event in the history of thought and the sociology of economics — a standard mathematical result which is simply wrong. It is especially interesting as the proof that Judd made a whoopsie was published years ago, yet the false alleged result survives. One might almost suspect that ideology or class interest is involved.

The key issue is that Judd considered tax rates which change over time and their incentive effects and then casually assumed that the public sector budget is always balanced. I guess he guessed this was OK because of Ricardian equivalence which says that. given a long list of implausible assumptions, the timing of *lump sum* taxes doesn’t matter, so the timing of taxes only matters because of incentive effects.

In fact Judd’s alleged proof is completely invalid. It is simply a math mistake.

The model
There are 2 groups workers and investors. The workers consume all of their income which consists of a wage and, possibly, a subsidy from the state. Investors have capital income — interest after tax A_tf'(K_t)-tau_tK where tau_t is the rate of taxation of capital, A_t is their wealth and K_t is total capital (these must be equal under Judd’s assumption that the state neither borrows nor accumulates a sovereign wealth fund).

Investors maximize an intertemporal utility function with rate of imaptience rho. The claim is that if the state wishes to maximize a weighted average of workers’ instantaneous utility and investors’ instantaneous utility and also has rate of time preference rho, then Tau_t goes to zero as t goes to infinity.

Now first note that even if Judd were right it would tell us nothing about what taxes will be optimal for the next million years. Oddly, many people some of whom are economists (one of whom Edward Prescott has won the Nobel memorial prize in economics) conclude that taxes on capital income should be cut to zero right now.

Second allow the state to accumulate wealth and consider the simplest case in which investors maximize the discounted stream of the logarithm of their consumption. This means that they consume (rho)A_t no matter what Tau_t is. Assume that tau_t can’t be greater than some limit taumax or the state will grab capital instantly which is, in effect, a lump sum tax and doesn’t distort.

In this case, Tau_t does go to zero, because the state accumulates until it’s income covers all its expenses plus whatever subsidy it chooses to pay workers and the distribution of income is exactly that which it finds optimal and then ceases to tax as there is no reason to tax anyone. Note that in this case there is no trade off between efficiency and desired redistribution — the distribution converges to that desired as if there were no problems with incentives. This is roughly the opposite of the standard interpretation. In the long run, the distribution of income is exactly as desired. There is no more taxation because there is no more reason to tax.

(More generally if the elasticity of substitution is less than one in the model (as it is in the data) the state will redistribute more until the investors are relatively poorer than is optimal. This is because the income effect of the tax is greater than the substitution effect, so high taxes on capital income promote saving. But I want to mainly stick with logarithmic utility).

Now consider an extreme case in which the state cares only about the welfare of workers. Judd claims the result holds even in this case. He is wrong even if the state is allowed to accumulate a sovereign wealth fund. In that case, there is a loss due to investors consumption equal to (rho)A_t and the state aims to mimimize A_t. If there is an upper limit on Tau, then Tau_t is always at this limit. The optimal policy is to tax capital income at the maximum rate allowed forever.

Now consider Judd’s assumption that the state can’t accumulate a sovereign wealth fund. The fact that it must leave the wealth in the hands of investors who consume it at rate rho means the optimal steady state K=K* is what it would be if there were depreciation at rate rho. This means that f'(K*)=2rho.
For investors to choose this steady state, it must be that the after tax return on capital (f'(K*) – tau) is equal to rho so 2rho -tau = rho and tau=rho.

So where did Judd go wrong ? His alleged proof included the assumption that the economy would converge to a steady state. He assumed that A = K, but also considered only the social budget constraint and concluded that, in the optimal steady state f'(K) = rho (as would be true if the state had access to optimal non distortionary taxation or if it could accumulate a sovereign wealth fund). But the restriction A=K is binding & it has a non zero shadow price. That shadow price is not constant even if all observables K, A, w, R consumption of capitalists and consumption of workers are constant.

Given the problem as stated, the economy can’t reach a steady state. The gain to the social planner of being able to accumulate wealth becomes constant. It’s current value grows at rate rho.

The impressive thing is that the alleged result is still accepted even though the proof that it is a math mistake was published over a decade ago.