# Summary of “A Tale of Three Elasticities”

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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…