WHY THE OPTIMAL LONG-RUN TAX RATE ON CAPITAL IS ZERO ... OR VERY HIGH: THE MISSING EXPLANATION.

• Author: Sanchirico, Chris William

I.Introduction 218 II.Basic Explanation 232 A. Step 1: Testing an Optimum With Ever Later Single-Period Down-Pulses in the After-Tax Return to Savings 233 B. Step 2: Dividing the Social Welfare Impact of a Single-Period Down-Pulse Into Three Parts 234 C. Step 3: Comparison of Savings Responses Across the Two Model Variants 235 1. Lead-Up Periods 236 2. Follow-On Periods 238 D. Step 4: Basic Engine--Infinity Paradoxes 238 E. Step 5: The Multiplicative Filters That Translate Savings Responses into Marginal Social Welfare 240 F Step 6: Two Ways to Use the Multiplicative Filters 242 G. Step 7: Application to OJM variant 242 H. Step 8: Application to SW Variant 243 III.Why the Pile Metaphor is Apt For Unfiltered, Socially Discounted Savings Responses 247 A. Counting Savings Responses After Socially Discounting Them/Grossing Them Up to the Down-Pulse Period 248 B. Lead-Up Periods 249 1. Legacy Lead-Up Periods 250 2. Newly Added Lead-Up Period 252 C. Follow-On Periods 254 D. Summary 254 IV.More on the Multiplicative Filter 255 V.Why Sending MPK-ATGR to Zero Does Not Work in the SW Variant 257 A. Add-On Perturbations Producing an Augmented Down-Pulse 258 B. Lead-Up Effect of Augmented Perturbation Grows Toward Positive Infinity as Initial Down-Pulse is Pushed Ever Farther Into Future 261 C. Fixed Behavior and Follow-On Effects 263 D. Summary and Implications for Lead-Up, Fixed Behavior and Follow-On Effects 265 E. Implications for Multiplicative Filter 266 F Aside: How the Add-On Down-Pulses are Generally Constructed 266 VI.Proving or Assuming Convergence 269 VII.A Note on Differences in Savings Responses Across Model Variants 270 VIII. Conclusion 271 I. INTRODUCTION

The question of how and whether to tax capital income and wealth places in relief the tradeoff between equity and growth. On one hand, capital ownership is relatively concentrated, so that, all else the same, taxing capital is a powerful lever for progressivity. On the other, capital accumulation expands economic activity and boosts labor productivity, raising employment and wages. If taxing capital slows capital accumulation, the effect could be detrimental across the economic spectrum.

The theoretical economic literature on optimal capital taxation is a natural place to look for insight into these competing considerations. This Article analyzes one p rticular thread of that literature, the "Judd model." (1) In an often-cited article from the mid-1980s Judd presents the finding that the long-run tax rate on capital should be zero even if society places substantial weight on equity. Very recently Straub and Werning identify an implicit assumption in Judd's analysis and show how removing it produces contrary findings within the same model. (2) This Article sets out to provide a clear and accessible explanation of the results obtained in both model variants. That explanation in hand, it argues that the Judd model has little to offer on the practical question of how optimally to tax apital.

The Judd model is populated with three kinds of actors who interact over an infinite number of periods. First, "capitalists"--who are all identical, infinitely lived and rational maximizers of their own intertemporal utility--enter the model endowed with all of the economy's resources and decide how to allocate those resources to their consumption in each period, over all periods stretching into the endless future. They make that decision based on perfect foresight regarding the infinite sequence of after-tax rates of return that they will be able to earn on their savings in each period. The resources they save in any period are--by an offstage market process--combined with labor to produce output in the immediately following period, out of which, inter alia, pretax returns are p id to the capitalist. Market forces are assumed to equate the pre-tax rate of return to the marginal product of capital. (3) Capitalists supply no labor. (4)

Second, "laborers"--who are identical and essentially inanimate--supply a fixed amount of labor in every period and always fully consume the sum of their wages and government transfers; they never save.

Third, an omniscient "government," before time begins, chooses, announces and credibly commits to the sequence of tax rates on capital that will apply for all time. It does so taking into account how this sequence will affect the prospective trajectory of capitalists' savings levels. Labor income is not taxed. The government cannot save or borrow, and in every period it must set aside a p rticular time-constant amount of revenue for non-transfer spending.

The tradeoff between equity and growth is starkly represented in the model. The government's policy objective is purposefully extreme to highlight the equity side of the balance: it cares only about the utility of the laborers. The growth counterweight is manifest in the government's chief constraint: the only way to increase laborers' consumption is to inspire capitalists to save more, and the only way to accomplish that is through altering the tax) n savings.

Greater capitalist savings leads to greater laborer consumption through two avenues. First, the increase in productive capital increases labor productivity and thereby the wage; the wage rate is assumed to equal the marginal product of labor by virtue of offstage market forces. Second, when savings increases, the tax base for capital taxation increases. This increases tax revenue and thus also increases transfers to laborers. (5)

In the context of this model--obviously artificial, but potentially informative--the following question is posed: how should the government set the infinite trajectory of tax rates on capital? Judd's original finding answers that question in part. The precise optimal infinite sequence of capital taxes is not determined, dependent as it is on unknowable p rameter values that are left as variables in the model. Rather Judd's finding characterizes what the sequence must eventually look like whatever values such p rameters take. In p rticular, Judd presents the result that the tax rate on capital must "converge to zero" over time. That is, the following must be true of the optimal sequence of tax rates: given any "collar" around zero, however tight, there is a point in time, however distant, after which the optimal sequence of tax rates stays within that collar.

Judd's article was, as noted, published in the mid 1980s. Over the course of the ensuing thirty years--through the restoration of preferential capital gains rates in the 1990s; (6) through the reduction of those preferential rates, their extension to dividends and the rollback of estate and gift taxation in the 2000s (7); through to the end of the 2010s--Judd's finding remained an important part of the brief for lowering taxes on capital.

But there was always a nagging puzzle. The Judd model's Economics 501-level result did not seem to jibe with a rudimentary lesson from Economics 101. In the basic model of individual choice there is no general prediction that taxing the return to savings reduces savings. This ambiguity is not a matter of theoretical oddities that may be safely disregarded. It arises from something as elemental as the difference between how many units are purchased and how much is spent on those units.

It is generally understood that even though increasing the price of gasoline will cause individuals to decrease how many gallons of gas they buy (or so we may safely assume), this does not mean that it causes them to spend less on gas. What happens to spending is a race between buying fewer gallons and paying more for each. Spending on gas--which is not quantity Q, but PxQ, where P is price--decreases if and only if the percentage reduction in Q is greater than the percentage increase in P. In other words, spending decreases if and only if gasoline is sufficiently "price elastic."

Increasing the tax rate on the return to savings effectively increases the price of future consumption (or other future uses like bequests) in terms of forgone current consumption. If ATGR is the aftertax gross rate of return on savings--"gross" in the sense that it includes the return of princip 1--each unit of current consumption that is forgone enables ATGR more future consumption. Reciprocally, to "purchase" a single unit of future consumption one must "p y" P = 1/ATGR units of current consumption. A tax based on capital income or ownership lowers ATGR and so raises P = 1/ATGR. Even though it may be safe to assume that the individual "buys" less future consumption in response, this does not necessarily imply that she "spends" less on future consumption. Savings may be regarded as the amount of current consumption that is "spent on" future consumption. Savings (P x Q) decreases if and only if the percentage decrease in future consumption (Q) is greater than the percentage increase in its price (P = 1/ATGR)--that is, if and only if future consumption is sufficiently price elastic. (8)

The puzzle for Judd's zero-tax result was how it accommodated the possibility that taxing the return to savings might actually increase savings. How could it be true, one might have asked, that raising the long-run tax rate above zero never improves social welfare even though it might well increase savings, productive capital, wages and government transfers to labor all the way back to the beginning of time?

It would have been one thing had Judd been ruling out the possibility that savings would increase in response to greater taxes based on an assessment of the empirical literature on actual savings behavior. To be sure, there has never been clear empirical evidence regarding the elasticity of future consumption (not to mention other future uses like bequests). (9) But even if the evidence had always been clear, the puzzle would remain that Judd's zero-tax result appears to apply without restriction. The Judd model makes no assumption about the price elasticity of...