Summary of “A Tale of Three Elasticities”

9 min readJul 30, 2020

Today I’ll be providing an overview of one of my favorite studies, “Optimal Taxation of Labor Incomes: A Tale of Three Elasticities” by Piketty, Saez, and Stantcheva, linked here. Everything here is as presented in the paper, except for my own notes, which I will put in brackets [like this]. Alright, let’s get started!

1. Introduction

This study focuses on the optimal marginal tax rate on the labor incomes of the top 1%, a very relevant topic today. It’s an empirical fact that since the 80’s, top tax rates have gone down, while the 1%’s share of national income has gone up. There are several explanations of this. First is simply the supply-side scenario: when you tax people less, they work and earn more. Second is the tax avoidance scenario: the idea is, the rich have always been making this much money. But now that tax rates are lower, they have less reason to hide their income. This certainly is a big part of short and even medium term responses to taxation, but we’re looking at the long term here. Third, perhaps with lower tax rates, the rich found it was worth their while to bargain for higher pay, knowing it wouldn’t just be taxed away. This higher pay would of course come at the cost of other groups. The goal of this paper is to assign an empirical value to each of these explanations using the elasticity of top taxable incomes to the net-of-tax rate (the percentage of income you take home after taxes, abbreviated ETI). For reference, the overall ETI is denoted e. An elasticity just measures the percent change in one variable over the percentage change in another. So for example, if e = .1, that means a 10% increase in taxes leads to a 1% decrease in taxable income.

2. Real Responses

First, the authors derive a model to measure the supply-side elasticity, or e₁. They also call this the real elasticity, since it is based on the responses of individuals and not on tax systems or institutions [an assumption I’d like to examine at some future date]. Higher taxes raise the “cost” of working in terms of effort, so higher taxes on the rich mean they will work less (perhaps less hours or years, or they won’t try to get a higher paying job). The authors go into a formal model of this, but I’ll just give the result. The revenue-maximizing tax rate with only a supply-side elasticity is:

Where a is the Pareto parameter (basically a measure of the distribution of income). I’ll include the proof at the end of the article, if you’re curious. One important note is that normally, the optimal rate is lower than the revenue-maximizing rate. This is because taking tax money from people hurts them, meaning higher government revenue has a cost as well as a gain. But for the top 1%, we can safely assume they aren’t hurt by higher taxes, since they already have plenty of money. So, the optimal and revenue-maximizing rates are the same.

3. Tax Avoidance

Next, the authors add tax avoidance to the model, measured by the elasticity e₂. Beyond the illegal, there are lots of legal ways to dodge taxes, like off-shoring, using tax expenditures (deductions, credits, and exemptions), disguising what is really income as something else (a “business trip” that’s really a vacation), and perhaps most importantly income shifting. Income shifting is when you disguise what should be labor income as capital income to take advantage of the lower rates on the latter. Tax avoidance again carries a cost, not just because it takes effort, but because “hidden income” like the business trip above are probably worth less to the rich than just having cash to do what you want with. Again, the authors develop a formal model for this, but the conclusion is the optimal and revenue-maximizing top tax rate in a model with supply-side effects and tax avoidance is:

Where t is assumed to be constant, and is the lower tax rate income faces after whichever method of avoidance is used. However, it of course maximizes government revenue if you crack down on tax dodging. Quoting the paper:

While the government can’t change underlying individual preferences and hence the size of the real elasticity, it can change the tax system to lower avoidance opportunities.

This is not to mention that tax dodging distorts economic activity. But fixing tax evasion can range from simple to impossible, depending on the source. Something like taxing the interest from municipal bonds is easy, while repealing the mortgage interest deduction is politically hard, taxing offshore accounts requires an unheard of level of international co-operation, etc. Meanwhile, small, informal cash businesses are basically impossible to tax in practice. [This is why formalizing transactions is key to a good tax system, by the way!] This all of course assumes that the tax avoidance is wasteful [something I intend to examine in-depth with regard to tax expenditures]. They develop a model to specifically account for the fact that capital and labor income will be concentrated differently, and may have different real elasticities. [I’ll examine this model later, as part of a full article on the issue of capital income taxation.]

4. Compensation Bargaining

In the classic perfectly competitive model for labor markets, your wage is equal to what you add to production, otherwise known as your marginal product. But lots of evidence shows “pay is not equal to marginal product for top earners”, like how pay goes up when the economy is booming, even if the company isn’t. Why? Well for one, executives exercise some level of power over the boards that set their pay. For another, we don’t live in some perfect world where everyone finds the best job that fits them immediately. In other words, there are frictions in the matching process between the supply and demand for jobs. And frictions mean inefficiencies, or rents (to use the economics term). “[T]he wage rate is not pinned down and can actually be anywhere in a band bounded by the outside options of the employer and the employee”. Wage bargaining is what determines who gets these rents. Bargaining, of course, takes effort. As top taxes go down, it’s more worthwhile to the rich to bargain harder. But this has to come at the cost of someone, either shareholders (remember we are talking about top LABOR incomes here), other workers, or consumers (through higher prices). In this model, the authors make the simplifying assumption that top bargaining comes at the cost of everyone equally. [I would say this assumption is a bit rosy: it likely comes at the cost of the poor far more than the upper-middle class.] Now since bargaining just changes the distribution of income, rather than the amount of income like the supply-side elasticity does, it’s assumed the government can tax away any redistributed bargaining rents fully. With a bargaining elasticity of e₃, this puts the revenue-maximizing rate at:

As you can see, as e₃ approaches e, the revenue-maximizing rate approaches 100%. The magnitude and sign of e₃ and top earner overpay can change the optimal formula considerably, putting it anywhere between 40% and 85%.

We can also put together all of the elasticities in the formula below, assuming e = e₁ + e₂ + e₃:

5. Empirical Evidence

The authors now turn to estimating the parameters discussed above, looking at US data. Income shifting is thought to be the main channel of tax avoidance, and we turn to that first.

In the short-run, to be sure, there is strong evidence … of large tax avoidance responses in various tax reform episodes with clear differential responses for top incomes including vs. excluding realized capital gains. But in the long run, the income shifting elasticity e₂ (as estimated long the ordinary income vs capital gains margin) appears to be small (say, e₂ < 0.1).

As mentioned earlier, offshoring and employee perks could be avenues of tax avoidance. But these have only increased while top income tax rates have decreased, the exact opposite of what we would expect if higher tax rates encouraged avoidance in this channel. Importantly, the authors do not attempt to construct estimates of broader income, which would include not just capital gains and taxable labor income, but various tax expenditures as well. [As I have mentioned before, this is important to include].

Next, they find that there is basically 0 correlation between total national income and top tax rates. This suggests e₁ is small, because if it was large, the rich working less would significantly detract from national income. There is also a significant positive correlation between the top tax rate and the income of the bottom 99%, suggesting a large e₃. From this, it’s tempting to say e ≈ e₃. But “evidence based on a single country is at best suggestive. Hence, to make further progress, we now turn to international evidence.”

This evidence includes data from 18 OECD countries going back to at least the 70’s. Top income shares varied a lot at the time, from 5% of total income in Denmark and Sweden to 10% in Germany. By 2004–2008, the range was from Denmark at 4% to the US at 18%. It’s notable that “no country experiences a significant increase in top income shares without implementing significant top rate tax cuts.” They estimate e to be anywhere from 0.2–1 depending on the country, and 0.5 overall. This range suggests that institutions, from unions to tax and spending structure, influence the elasticity a lot. It’s certainly true that both make it harder and less worthwhile to bargain. The explanation that increased top income shares are a result of a skills gap between top earners and the rest is rejected, as it can’t explain the large range of top income shares, or the correlation with top tax rates. They also find that e₁ is definitely ≤ 0.2, again meaning a large role for bargaining. This checks out with other evidence: the rich haven’t really worked more or retired earlier since top tax rates were lowered, suggesting a small labor supply-side response.

6. Conclusion

In conclusion, the authors note that more research is needed to determine how much the rich are overpaid. This changes how much of the total elasticity comes from bargaining, rather than supply-side effects. As mentioned, all of these formulas assume no weight is given to money earned by the top 1%. [I would argue this isn’t strong enough: at least for the very rich, we should give them somewhat negative weights, because they rent-seek and corrupt democracy.] Lastly, here are a collection of empirical estimates based on different scenarios. Note that some assumptions are required for e to equal e₁ + e₂ + e₃.