metadata: authors: 0: author: Bergstrom,Katy Ann
1: author: Dodds,William
count: World
docna: 0: docna: Using Labor Supply Elasticities to Learn
            about Income Inequality : The Role of Productivities versus Preferences
docty: Policy Research Working Paper
subtopic: Poverty Reduction Strategies,Public Sector Economics,Public Finance Decentralization and
            Poverty Reduction,Macro-Fiscal Policy,Taxation & Subsidies,Economic Adjustment and Lending,Rural Labor Markets,Labor Markets
teratopic: Poverty Reduction,Public Sector Development,Macroeconomics and Economic Growth,Rural Development,Social Protections and Labor
abstracts: cdata!: This paper argues that labor supply
            elasticities encode information about the determinants of
            income inequality. In the theoretical framework, individuals
            choose labor supply conditional on productivities and
            preferences for consumption relative to leisure. The paper
            shows that reduced-form labor supply elasticities allow one
            to isolate the components of income due to productivities
            versus preferences. The paper then investigates what labor
            supply elasticities imply about the importance of
            productivities versus preferences in the United States.
            Estimates from the literature imply productivities drive
            most of income inequality. Larger income effects and larger
            differences between income and hours worked elasticities
            imply preferences play an increasingly important role.
repnme: repnme: Using Labor Supply Elasticities to Learn
            about Income Inequality : The Role of Productivities versus Preferences
display_title: Using Labor Supply Elasticities to
            Learn about Income Inequality : The Role of Productivities
            versus Preferences
listing_relative_url: /research/2020/01/31683966/using-labor-supply-elasticities-learn-income-inequality-role-productivities-versus-preferences
topicv3: Education, Skills Development and Labor
            Market m1326135 1484,Labor Markets m1326488 855,Taxation m1327615 1121,Income Inequality m1503891 2668,Labor Supply m1504055 2751
publishtoextweb_dt: 2020-01-06T00:00:00Z
historic_topic: Poverty Reduction,Public Sector Development,Macroeconomics and Economic Growth,Rural Development,Social Protections and Labor
origu: DECRG: Poverty & Inequality (DECPI)
no_of_pages: 73
author_keys: 0: author_key: 1811674
model_extractions: Policy Research Working Paper 9102 Using Labor Supply Elasticities to Learn about Income Inequality The Role of Productivities versus Preferences _Katy Bergstrom William Dodds_ Development Economics Development Research Group January 2020
Policy Research Working Paper 9102

# **Abstract**

This paper argues that labor supply elasticities encode information about the determinants of income inequality. In the theoretical framework, individuals choose labor supply conditional on productivities and preferences for consumption relative to leisure. The paper shows that reduced-form labor supply elasticities allow one to isolate the components of income due to productivities versus preferences. The paper then investigates what labor supply elasticities imply about the importance of productivities versus preferences in the United States. Estimates from the literature imply productivities drive most of income inequality. Larger income effects and larger differences between income and hours worked elasticities imply preferences play an increasingly important role.

This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at kbergstrom@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Using Labor Supply Elasticities To Learn About Income Inequality: The Role of Productivities versus Preferences<sup>_∗_</sup> Katy Bergstrom<sup>_†_</sup> William Dodds<sup>_‡_</sup> _Keywords: income inequality, productivity, preference, labor supply elasticity JEL Codes: D63, J22, H21, H31_

> _∗_ We would like to thank our advisors, Doug Bernheim, Raj Chetty, and Caroline Hoxby for their guidance and support on this project. We would also like to thank Jose Maria Barrero, Pascaline Dupas, Petra Persson, Alessandra Peter, Luigi Pistaferri, Juan Rios, Florian Scheuer, and Isaac Sorkin for helpful advice, as well as participants at various seminars at Stanford University for their useful comments. Finally, thanks to The Ric Weiland Graduate Fellowship in the School of Humanities and Sciences, the B.F. Haley and E.S. Shaw Fellowship for Economics, and the Arthur and Eva Karasz Fellowship for financial support.

> _†_ Development Research Group, World Bank. Email: kbergstrom@worldbank.org.

> _‡_ Charles River Associates. Email: wdodds@crai.com.
# **1 Introduction**

The determinants of income inequality are a contentious topic of debate among economists, politicians, and policy makers alike. Many factors, such as family background, demographics, discrimination, genetics, luck, and work ethic play a role in determining economic success. All of these factors ultimately contribute to two higher level determinants of labor income inequality: (1) differences in productivities, i.e., ability to transform labor into personal income, and (2) differences in preferences, i.e., desire for consumption relative to leisure.<sup>1</sup> Note our definition of preferences is narrow in that it refers only to the taste for consumption relative to leisure whereas our definition of productivity is broad in that it encompasses not only things like intelligence or social skills but also things like human capital acquisition, discrimination, or rent seeking, which all ultimately impact ones ability to transform labor into income.

Understanding the determinants of income inequality is important because social welfare gains associated with redistribution depend on _why_ we have inequality. A number of studies have shown that individuals’ redistributive tastes may be influenced by how much of income inequality is driven by preferences. In experimental settings, individuals appear less inclined toward redistribution when income differences are due to differences in preferences, which manifests as differential effort (e.g., Hoffman et al. (1994), Cherry et al. (2002), or Rey-Biel et al. (2011)). Moreover, Alesina et al. (2001) find that countries in which individuals believe income is driven primarily by preference differences tend to have less redistributive policies. Hence, individuals’ _normative_ tastes for redistribution appear to depend on the sources of income inequality; thus, examining the extent to which income inequality is driven by productivity vs. preference heterogeneity is a _positive_ step towards understanding the welfare benefits of redistribution.<sup>2</sup> This paper shows that we can use information encoded in labor supply elasticities to learn about the extent to which labor income inequality is driven by heterogeneity in productivities vs. heterogeneity in preferences for consumption relative to leisure. To see why labor supply elasticities contain information about income inequality, consider a population of individuals in a static world who choose how many hours to work conditional on heterogeneous productivities and preferences over the consumption/leisure trade-off. To simplify ideas, assume productivity is equivalent to the hourly wage rate. In this canonical world, labor income inequality can first be decomposed into differences in hourly

> 1We will focus solely on labor income inequality throughout this paper.

> 2However, decomposing income inequality into productivities and preferences is not necessarily complete; if individuals have redistributive preferences which depend on the determinants of productivities and preferences, then a more complete decomposition of income inequality will be needed to assess the welfare benefits of redistribution. For example, individuals may have different tastes for redistribution if preference heterogeneity is mostly due to innate disutility of labor vs. disutility of labor due to poor health. Nonetheless, even if this is the case, our work is a useful step towards better understanding the sources of income inequality.

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wages and differences in hours worked (such a decomposition is explored in, for example, Haider, 2001 or Blundell et al., 2018). Going a step further, heterogeneity in hours worked is driven not only by differences in preferences over the labor/leisure trade-off, but also by differences in wages (productivities), which lead to gross substitution effects as higher wage individuals shift toward labor (or toward leisure if the labor supply curve is backward bending). The labor supply elasticity of hours worked with respect to the wage rate tells us how we expect hours worked to change with the wage rate, holding preferences over the consumption/leisure trade-off constant. Thus, we can use the labor supply elasticity to net out the component of hours worked due to wage effects, leaving us with the component of hours worked attributable to preference differences. Finally, we can use the component of hours worked attributable to preferences to understand how much of income inequality is due to preference heterogeneity.

We formalize this intuition by developing a method to assess the extent to which crosssectional labor income inequality is driven by productivities vs. preferences, using only empirically observable labor supply elasticities, in the context of a general labor supply model. To begin, we consider a static neo-classical labor supply model in which productivity is equivalent to the hourly wage; we later relax this assumption. Preferences are captured by a single parameter that changes the marginal rate of substitution between consumption and leisure. Conditional on heterogeneous productivities and preferences, agents choose hours worked to maximize utility. This agent optimization problem defines a function between primitives (productivities and preferences) and labor supply decisions (incomes and hours worked). We show how to invert this function, i.e., how to infer primitives from observable decision variables; this allows us to investigate the role productivities and preferences play in driving income inequality. The principle finding is that we can express this inverse function entirely in terms of reduced form labor supply elasticities, thereby yielding a transparent procedure that highlights the manner in which labor supply elasticities encode information about the drivers of income inequality. Towards understanding this result, consider comparing preferences for consumption relative to leisure between a high wage individual who works many hours and a low wage individual who works fewer hours. The labor supply elasticity tells us how hours worked should vary between a high wage person and a low wage person, holding preferences constant. Thus, we can use the labor supply elasticity to subtract out the component of hours worked due to wage effects and then use the component of hours worked solely due to preferences to compare preferences between the high wage person and the low wage person. Essentially, labor supply elasticities encode information about income inequality as they allow us to infer individual preferences from the component of hours worked that is _not_ driven by wage differences.

Our baseline model assumes that all individuals are able to optimally adjust their income on the intensive margin. We show that labor supply elasticities still contain 2
information about income inequality even if individuals face labor market frictions so that labor supply is not perfectly flexible. Our method can be adapted to determine the extent to which income inequality is driven by productivities (hourly wages), preferences, and frictions if we can elicit how much individuals would ideally like to work (for example, via survey) and the elasticity of ideal hours worked with respect to the wage rate.<sup>3</sup> We then show that labor supply elasticities still encode information about income inequality if we relax the assumption that productivity is equivalent to the hourly wage. We consider a world in which individuals choose (unobserved) effort per hour in addition to hours of work. In this setup, hourly wage is equal to effort per hour multiplied by productivity (thus, productivity is now equal to an unobservable _effort_ wage). Our main result is that we can still invert the function between productivities and preferences and incomes and hours worked as long as we observe the labor supply elasticities of _both_ income and hours worked with respect to the effort wage, or equivalently the tax rate. We can do this inversion because the function mapping productivities and preferences to labor supply decisions has a derivative matrix that can be expressed entirely in terms of observable labor supply elasticities; hence, we can invert this observable matrix to find the derivative matrix of the inverse function, which in turn allows us to recover the entire inverse function (up to an irrelevant normalization). Thus, our method allows us to investigate the extent to which productivities (effort wages) and preferences impact income inequality using only four estimable labor supply elasticities: the taxable income and hours worked elasticities, both compensated and uncompensated, with respect to the tax rate.<sup>4</sup> We then show that labor supply elasticities still contain information about income inequality even if productivity (the rate at which one transforms labor into personal income) is partially determined by prior human capital decisions. We explore a dynamic labor supply model with human capital acquisition and endogenous wage growth. We show that we can use labor supply elasticities to invert the relationship between observables (incomes and hours) and _current_ productivities and preferences, recognizing that current productivities are determined both by innate skills as well as past labor supply and human capital decisions (which are choice variables and therefore functions of both innate skills and preferences). Thus, labor supply elasticities allow us to recover _crosssectional_ productivities and preferences using labor supply elasticities. Intuitively, this yields a lower bound as to the extent that preferences drive income inequality because some of the variation in productivities may actually be driven by previous human capital decisions, which are in turn partly driven by preferences.

Next, we illustrate what different labor supply elasticities imply about the drivers of

> 3While actually identifying the relevant elasticity of ideal hours worked with respect to the wage rate may be challenging, we discuss in the text two potential, yet imperfect, ways to get at this elasticity.

> 4The fully general version of our method allows labor supply elasticities to vary across individuals and requires elasticity estimates that vary at the income and hours worked level.

3
income inequality in the U.S. Using several labor supply elasticity estimates and using data on incomes and hours worked from the American Time Use Survey, we apply our method to recover individual productivities and preferences.<sup>5</sup> We begin with a baseline set of elasticity parameters taken as averages from a number of labor supply studies discussed in Chetty (2012). Under the baseline elasticities, we infer high income people actually have lower average preferences for consumption compared to middle income people. Essentially, the baseline labor supply elasticities imply that high income people should work substantially more than low income individuals; because we observe a relatively flat hours gradient over the income distribution, we infer that high income people have, on average, lower preferences for consumption relative to leisure. However, we then vary the elasticity parameters and show that we will infer preferences are increasingly important in driving income inequality if we use (1) a larger difference between the income and hours elasticities and/or (2) larger income effects to recover individual productivities and preferences. Thus, a larger difference between the income and hours elasticities and larger income effects imply that more of the difference in incomes between rich and poor is due to preferences.

Finally, to highlight how our findings on the determinants of income inequality change our understanding of the welfare benefits of redistribution, we simulate optimal tax schedules that account for both productivity and preference heterogeneity driving inequality. While the general methodology to recover determinants of income inequality devised in this paper is free of normative assumptions, for the purpose of welfare calculations, we adopt the normative stance developed in Fleurbaey and Maniquet (2006) in which differences in productivities merit redistribution whereas differences in preferences do not merit redistribution. We simulate optimal tax schedules, accounting for both productivity and preference heterogeneity, under different values of the relevant labor supply elasticities and compare these schedules to the optimal tax schedules in which all income inequality is due to productivity heterogeneity as in Mirrlees (1971) or Saez (2001).

Under our baseline elasticity estimates from Chetty (2012), we find that optimal tax rates are actually slightly higher than the Mirrleesian reference case in which all income inequality is due to productivity heterogeneity. Essentially, the elasticity estimates from Chetty (2012) imply that high income individuals actually have lower preferences for consumption, on average, than middle income individuals. Therefore, high taxes are even more desirable than in the Mirrleesian benchmark. However, this finding is sensitive to the elasticity estimates used to recover individual productivities and preferences. We find that a larger difference between the income and hours worked elasticities and larger income effects both imply lower tax rates relative to the Mirrleesian optimal tax schedule.

> 5Because the literature has not reached a consensus on the magnitudes of the different labor supply elasticities, and because current data provides imperfect measurements of hours worked, our empirical application aims to investigate how changing labor supply elasticities affects the relative importance of productivities vs. preferences in driving income inequality.

4
This is because a larger difference between the income and hours worked elasticities and larger income effects both imply that high income individuals have higher preferences for consumption, so that redistributing away from them is less desirable than in the Mirrleesian benchmark. The takeaway is that labor supply elasticities not only impact the efficiency costs of taxation, but also encode important information about the equity benefits of taxation through the information they contain about the drivers of income inequality.

The rest of the paper proceeds as follows: Section 2 discusses related literature, Section 3 illustrates how labor supply elasticities can be used to recover preferences in the context of a labor supply model in which productivity is equivalent to the hourly wage and individuals only choose hours worked, Section 4 extends our analysis when productivity is not equal to the hourly wage (i.e., individuals choose effort per hour as well as hours worked), Section 5 discusses our empirical implementation, Section 6 discusses how our findings impact optimal taxation, and Section 7 concludes.

# **2 Related Literature**

This paper is related to four different strands of literature: (1) decomposing income inequality empirically into wages and hours, (2) survey evidence on the determinants of income inequality, (3) the relationship between income inequality determinants and redistribution, and (4) the invertibility of economic systems.

First, this paper is related to an empirical literature focused on statistically decomposing income inequality into wage heterogeneity vs. hours worked heterogeneity. Haider (2001), for example, decomposes the variance of income into wage and hours worked, finding that most of income variance is due to wage variance, although a non-negligible amount of income variance is due to hours variance (and the covariance between hours and wages). Doiron and Barrett (1996) also find that income variance is driven more by wage heterogeneity for males; however, they find that the opposite is true for women. Gottschalk and Danziger (2005) and Blundell et al. (2018) have similar findings. Our paper contributes to this literature by going a step further and decomposing income inequality into productivity and preferences, which is the more relevant decomposition for welfare analysis; our decomposition recovers preferences from hours worked net of substitution effects and recognizes that productivity may not be equivalent to the hourly wage if individuals exert differential effort per hour.

Second, while, to the best of our knowledge, this is the first paper that attempts to decompose income inequality into productivity heterogeneity vs. preference heterogeneity, there is a large body of work that investigates individuals’ _beliefs_ over the determinants of income inequality and how these beliefs relate to views on the merits of redistribution. Data from the World Values Survey shows Americans are about twice as likely 5
as Europeans to think that the poor are lazy or lack willpower (60% versus 26%) and that in the long run, hard work usually brings a better life (59% versus 34-43%; Ladd and Bowman (2001)). Notably, the United States provides far less in welfare assistance than most European countries. This correlation between beliefs and actual redistributive policies across countries is reinforced by the findings of Alesina et al. (2001): social spending (welfare, social security, etc.) as a percentage of GDP is positively correlated with average beliefs that income inequality is driven by luck as opposed to preferences.

These cross-country findings are further supported by experimental evidence. For example, Hoffman et al. (1994) use an experiment to show that when agents earn the right to be the dictator, they give less in the dictator game. Similarly, Cherry et al. (2002) and Oxoby and Spraggon (2008) show that dictators give (take) less when income is earned by the dictators (recipients) compared to when income is determined by the experimenter. Investigating the role of beliefs on the causes of poverty and the differences in redistribution policy between Spain and the US, Rey-Biel et al. (2011) show that overall giving between American and Spanish subjects is similar when the actual role of luck versus effort is known, however, Spanish subjects give more when uninformed compared to American subjects because American subjects have stronger ex ante beliefs that effort is the primary driver. In summation, individuals look upon redistribution towards poorer individuals more favorably when these individuals are perceived to be poor due to luck as opposed to low preferences for consumption relative to leisure.

Third, there have been a number of papers which explore how determinants of income inequality affect optimal redistribution from a theoretical perspective. For example, Boadway et al. (2002), Chon ́e and Laroque (2010), Jacquet and Lehmann (2015), and Lockwood and Weinzierl (2016) all explore how adding an additional dimension of heterogeneity in the form of preferences affects different aspects of the tax schedule. For example, Lockwood and Weinzierl (2016) show that, under certain functional form assumptions, increasing the amount of income inequality due to preferences leads to less redistributive optimal tax schedules. Our work contributes to this literature by showing, via simulation, how different labor supply elasticity parameters change optimal tax schedules by impacting the welfare benefits of redistribution (through the implied degree of preference vs. productivity heterogeneity driving inequality).

Lastly, this paper is related to the literature on invertibility of economic systems. Saez (2001) shows in a labor supply model with productivity heterogeneity how one can invert labor supply decisions into productivities using the elastictiy of income with respect to the tax rate. We extend this result to a model with both preference and productivity heterogeneity. There is also a vast literature that performs inversions from observables into unobservables in the context of product demand using structural estimation, either parametrically (e.g., in Berry et al. (1995)) or non-parametrically (e.g., in Berry and Haile (2010)). Our method to invert observable labor supply decisions into unobserved 6
primitives uses economic theory to relate the elasticities of observables with respect to primitives to elasticities of observables with respect to other observables (in our case tax rates). This allows us to bypass structural estimation of utility functions by using reduced form elasticity estimates to directly invert the relationship between observables and primitives. Our method is useful conceptually towards understanding the relationship between primitives (productivities and preferences) and observable labor supply decisions. Moreover, our method is perhaps more transparent than a structural approach in revealing the variation driving the inversion; this is especially important in our labor supply context as it enables us to identify the particular features of the data that lead to the estimated income inequality decomposition.

# **3 Baseline Model**

We begin with a simple, static labor supply model with no labor market frictions in which productivity is equivalent to hourly wage. Individuals have only one dimension of labor supply: hours worked. We show that we can use labor supply elasticities to invert the relationship between observables and primitives in this stylized world. Because the insights of this baseline framework will carry over to more general cases, it is useful to highlight the underlying mechanisms in this simplified setting.

## **3.1 Problem Setup**

Suppose individuals have preferences over hours worked, _h_ , and consumption, _c_ , and that they vary in terms of their productivity, _n_ , and preferences for consumption relative to leisure, _α_ . Productivity affects the return to labor, with income _z_ = _nh_ , whereas preferences affect the marginal rate of substitution between consumption and leisure. Denoting the (linear) tax rate as _T_<sup>_′_</sup> and the guaranteed income level _R_ , the individual’s<sup>7</sup> problem can be written as:<sup>6</sup> The associated first order condition is given by:

> 6 We assume a linear tax rate first for expositional simplicity, but we consider piece-wise linear tax schedules with increasing marginal tax rates in Appendix A.4 in the context of the model in Section 4, which nests our baseline model.

> 7The assumption of additive separability is not necessary. Appendix A.5 shows how the analysis carries over to the non-separable case in the context of our more general baseline model with effort decisions.

7
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maps observables to primitives, entirely in terms of reduced form labor supply elasticities. Once we know _G_<sup>_−_1</sup> , we can find _G_ , which allows us to analyze the extent to which income inequality is due to differences in _n_ vs. differences in _α_ .

It is worth mentioning that an alternative route to determining _n_ and _α_ in the above model would be to make functional form assumptions on _u_ ( _·_ ) and _v_ ( _·_ ) (or parametrize these from data in some fashion) and use the individual first order condition to determine the value of _n_ and _α_ that would choose to optimally work hours _h_<sup>_∗_</sup> and earn income _z_<sup>_∗_</sup> . The key theoretical insight of this section is that we can instead use observable labor supply elasticities to recover _G_<sup>_−_1</sup> without making any functional form assumptions on _u_ ( _·_ ) or _v_ ( _·_ ). By expressing _G_<sup>_−_1</sup> in terms of a few observable elasticities, our sufficient statistics approach explicitly identifies how key economic parameters affect our inferences around the sources of income inequality. Before deriving _G_<sup>_−_1</sup> , we now take a short detour to define the relevant elasticity concepts.

# **3.2 Defining Labor Supply Elasticities**

The primary elasticities of interest in this paper are elasticities of choice variables, such as hours worked, with respect to productivities _n_ or preferences _α_ . While this section will focus on elasticities of hours worked, we define elasticities more generally as we will discuss elasticities of other choice variables (e.g., effort or income) later in subsequent sections. We define elasticities of choice variable _i_ w.r.t. _n_ and _α_ , respectively, as: We also define the uncompensated elasticity of a choice variable _i_ w.r.t. the tax rate as: We similarly define the income effect parameter as: _∂_ log( _i_<sup>_∗_</sup> <u>)</u> Finally, we define _ξi_<sup>_c≡_</sup> _∂_ log(1 _−T_<sup>_′_</sup> ) �� _c_<sup>asthecompensatedelasticity.BytheSlutskyEqua-</sup> tion, we have the following relationship:<sup>11</sup>

> 11 _∂_ log( _h_<sup>_∗_</sup> <u>)</u> _∂_ log( _h_<sup>_∗_</sup> <u>)</u> If the choice variable _i_ is hours worked, the Slutsky equation states: _∂_ log(1 _−T_<sup>_′_</sup> )<sup>=</sup> _∂_ log(1 _−T_<sup>_′_</sup> ) �� _c_<sup>+</sup> _n_ (1 _− T_<sup>_′_</sup> )<sup>_<u>∂h</u>_</sup> _∂R_<sup>_∗_.ThisisthestandardlaborsupplySlutskyequation(recognizingthat</sup><sup>_n_(1</sup><sup>_−T ′_)isthe</sup> _∂_ log( _h_<sup>_∗_</sup> <u>)</u> _∂_ log( _h_<sup>_∗_</sup> <u>)</u> _∂_ log( _h_<sup>_∗_</sup> <u>)</u> _∂_ log( _h_<sup>_∗_</sup> <u>)</u> after-tax wage, _∂_ log(1 _−T_<sup>_′_</sup> )<sup>=</sup> _∂_ log( _n_ (1 _−T_<sup>_′_</sup> ))<sup>and</sup> _∂_ log(1 _−T_<sup>_′_</sup> ) �� _c_<sup>=</sup> _∂_ log( _n_ (1 _−T_<sup>_′_</sup> )) �� _c_<sup>).Ifthechoicevariable</sup><sup>_i_is</sup> income _z_ , this Slutsky equation is the same as in Saez (2001).

9
Before we move on, note that in our labor supply model in which individuals only choose hours worked, the tax elasticities of incomes and hours worked are identical because agents have only one margin of adjustment: _ξz_<sup>_u_=</sup><sup>_ξ_</sup> _h_<sup>_u_and</sup><sup>_ξ_</sup> _z_<sup>_c_=</sup><sup>_ξ_</sup> _h_<sup>_c_.</sup>

# **3.3 Recovering Productivities and Preferences Using Labor Supply Elasticities**

We now proceed to derive the function _G_<sup>_−_1</sup> : _Z_<sup>_∗_</sup> _× H_<sup>_∗_</sup> _→ N × A_ in terms of labor supply elasticities. First, we can immediately recover each individual’s productivity _n_ as it is simply equal to the hourly wage; so if we observe incomes, _z_<sup>_∗_</sup> , and hours worked, _h_<sup>_∗_</sup> , we can recover _n_ = _z_<sup>_∗_</sup> _/h_<sup>_∗_</sup> . But how can we recover each person’s value of _α_ from _z_<sup>_∗_</sup> , _h_<sup>_∗_</sup> , and labor supply elasticities? The first step towards recovering preferences _α_ is understanding the relationship between hours worked and primitives. Hence, we state the following two Lemmas:

**Lemma 3.1.** _The elasticity of hours worked w.r.t. n, ξh_<sup>_n,isequaltotheuncompensated_</sup> _tax elasticity of hours worked, ξh_<sup>_u._</sup>

_Proof._ See Appendix A.1.

The intuition for Lemma 3.1 is that changing _n_ changes the relative price of leisure and generates an income effect. Similarly, changing the tax rate also leads to a change in the relative price of leisure as well as an income effect.

**Lemma 3.2.** _The elasticity of hours worked w.r.t. α, ξh_<sup>_α,isequaltothecompensatedtax_</sup> _elasticity of hours worked, ξh_<sup>_c._</sup>

_Proof._ See Appendix A.2.

The intuition behind Lemma 3.2 is that changing _α_ leads to a change in the price of leisure. Similarly, changing the tax rate also leads to a change in the price of leisure; however, changing the tax rate also leads to an income effect. Heuristically, changing _α_ leads to the same effect on hours worked as a change in the tax rate if we subtract out the income effect caused by the change in tax rates. But by the Slutsky equation, changing the price of leisure and subtracting out the income effect is the compensated elasticity; hence the elasticity of hours w.r.t. _α_ is the same as the compensated tax elasticity. Lemmas 3.1 and 3.2 yield the following two partial differential equations, respectively: 10
Note, _h_<sup>_∗_</sup> ( _n, α_ ) and _z_<sup>_∗_</sup> ( _n, α_ ) refer to the optimal incomes and hours chosen by individual ( _n, α_ ). To isolate the key economic ideas underlying the inversion between incomes and hours worked and primitives, let us assume that the tax elasticities _ξh_<sup>_u_and</sup><sup>_ξ_</sup> _h_<sup>_c_are constant.</sup> We will relax this assumption in the more general setup of Section 4. Under this constant elasticity assumption, we can trivially solve the system of partial differential equations given by 2 and 3, where _k_ is a constant: Using the fact that log( _n_ ) = log( _z_<sup>_∗_</sup> ) _−_ log( _h_<sup>_∗_</sup> ), we can solve for log( _α_ ) from Equation 4 by normalizing _k_ = 0 (which is without loss of generality as it just rescales preference parameters): But Equation 5 expresses _α_ in terms of _z_<sup>_∗_</sup> , _h_<sup>_∗_</sup> and labor supply elasticities. Hence, we have recovered the inverse function that maps incomes and hours back to primitives in our simple labor supply model with constant elasticities:

**Proposition 3.3.** _We can recover G_<sup>_−_1</sup> : _Z_<sup>_∗_</sup> _× H_<sup>_∗_</sup> _→ N × A from the elasticities ξh_<sup>_cand_</sup> _ξh_<sup>_uaslongasξ_</sup> _h_<sup>_c>_0</sup><sup>_:_</sup>

The key economic intuition behind Proposition 3.3 comes from the equation log( _α_ ) = log( _h_<sup>_∗_</sup> <u>)</u> _−ξh_<sup>_u_log(</sup><sup>_n_</sup><sup><u>)</u></sup> . The idea is that hours worked reveals information on preferences for con- _ξh_<sup>_<u>c</u>_</sup> sumption relative to leisure, but it is contaminated by substitution effects from different wage levels. In other words, we cannot directly infer preferences by examining hours worked because hours worked is a choice variable that depends on preferences as well as the hourly wage. In order to recover preferences for consumption relative to leisure we need to net out the component of labor supply due to wage effects; i.e., we need to determine hours worked conditional on having the same wage. As such, we subtract out the effect of wages on hours worked, _ξh_<sup>_u_log(</sup><sup>_n_), to determine the component of hours solely due</sup> to preferences, i.e., hours worked conditional on a common wage level: log( _h_<sup>_∗_</sup> ) _−ξh_<sup>_u_log(</sup><sup>_n_).</sup> Because hours worked is increasing with preferences, we can then compare this measure of hours conditional on a common wage, log( _h_<sup>_∗_</sup> ) _− ξh_<sup>_u_log(</sup><sup>_n_),torankindividuals’</sup> preferences for consumption relative to leisure. However, we can go a step further and 11
<!-- Start of picture text --> Cc mI)<br>Neng (1 —T’)<br>eng Qeng<br>i<br>/emech Nmecn(1 — T')<br>1\<br>\'<br>\ ;<br>h*(Mnecw Geng) h (Mechs Gnech) R Neng, % ng) h<br><!-- End of picture text -->
preferences between the engineer and the mechanic.

To recap, we have shown that we can recover individual preference parameters by inverting the function between primitives and labor supply decisions using just two observable labor supply elasticities. The key economic insight is that labor supply elasticities allow us to compare preferences for consumption relative to leisure across individuals with different hourly wages by subtracting out the component of hours worked due to wage effects. The component of hours worked _not_ due to wage effects (i.e., the component of hours worked attributable to preferences) then allows us to compare individual preferences.

# **3.4 Labor Supply Frictions**

Our results so far have relied on the assumption that individuals can perfectly optimize their labor supply by changing hours worked on the intensive margin. There is a non-negligible subset of people for whom this is probably a reasonable assumption (e.g., Uber drivers or the self-employed or commission workers); we can immediately apply our method to use labor supply elasticities to learn about the drivers of income inequality _within_ this subset of the population. On the other hand, many individuals do face labor demand inelasticity or market frictions; we provide an imperfect, yet easily implementable, modification to recover productivities and preferences using labor supply elasticities if individuals face frictions. Labor market frictions lead to two issues in applying our method: (1) individuals’ observed hours are no longer equivalent to their optimal hours and (2) elasticities of observed hours with respect to the tax rate, which reflect both labor market frictions as well as how optimal choices change with tax rates, are no longer equivalent to elasticities of optimal hours with respect to primitives (as in Lemmas 3.1 and 3.2). Towards broadening the applicability of our method, consider a world in which all individuals ( _n, α_ ) choose an optimal hours worked _h_<sup>_∗_</sup> ( _n, α_ ) yet face labor supply frictions. Thus, each individual ( _n, α_ ) ends up working _h_<sup> ̃</sup> ( _n, α_ ) = _h_<sup>_∗_</sup> ( _n, α_ ) + _εn,α_ , where _εn,α_ is some deviation from optimal hours (it does not matter what causes this deviation from optimal hours).

Even with frictions that prevent individuals from working their optimal number of hours, we can still learn about the role of productivities vs. preferences in driving income inequality from labor supply elasticities. To solve the first issue that observed hours are not equal to optimal hours, suppose that we were able to elicit (via survey, for example) individuals’ true optimal hours worked _h_<sup>_∗_</sup> .<sup>13</sup> Productivity is equal to observed income ̃ divided by observed hours, _n_ = _z/h_<sup> ̃</sup> , and optimal income is given by optimal hours _h_<sup>_∗_</sup> multiplied by _n_ : _z_<sup>_∗_</sup> = _nh_<sup>_∗_</sup> = ( ̃ _z/h_<sup> ̃</sup> ) _h_<sup>_∗_</sup> .

> 13For example, the National Study of the Changing Workforce asks people how many hours they would prefer to work.

13
There are at least two ways to deal with the second issue that observed hours elasticities are not equivalent to optimal hours elasticities. First, we could estimate, via survey, the elasticity of optimal hours worked to the tax rate (by asking individuals their preferred hours worked under their current wage before and after a tax change). This would allow us to directly recover _G_<sup>_−_1</sup> as in Proposition 3.3 using optimal hours _h_<sup>_∗_</sup> , optimal income _z_<sup>_∗_</sup> , and the elasticities of preferred hours. Alternatively, suppose that some known set of individuals have _εn,α_ = 0, so that their observed hours worked is equal to their optimal hours worked as they face no frictions (e.g., Uber drivers or the self-employed or commission workers). If we can estimate labor supply elasticities for this subset of individuals with _εn,α_ = 0, then we could recover _G_<sup>_−_1</sup> exactly as in Proposition 3.3 using optimal hours _h_<sup>_∗_</sup> , optimal income _z_<sup>_∗_</sup> , _ξh_<sup>_n_=</sup><sup>_ξ_</sup> _h_<sup>_u|ε_</sup> _n,α_<sup>=0and</sup><sup>_ξ_</sup> _h_<sup>_α_=</sup><sup>_ξ_</sup> _h_<sup>_c|ε_</sup> _n,α_<sup>=0.Suchaprocedure</sup> will allow us to determine the extent to which income inequality is driven by productivity heterogeneity, preference heterogeneity, _and_ frictions. While this subsection contains no additional theoretical insights beyond Proposition 3.3, we believe it may be useful for empirical applications of the method.

Next, we show that we can still use labor supply elasticities to learn about the drivers of income inequality even if we relax the assumption that productivity is equal to hourly wage.

# **4 What if Productivity Differs from Hourly Wage?**

While the baseline model in Section 3 is useful to conceptualize how preferences can be inferred from hours worked by subtracting out wage effects on hours worked, it abstracts from the possibility that productivity is not equivalent to the hourly wage. Why might productivity, the rate at which people transform labor into income, differ from the hourly wage? Previous studies, both theoretical and empirical, have stressed the importance of accounting for effort per hour decisions in labor supply models, e.g., Atkinson and Stiglitz (1976), Pencavel (1977), Lin (2003), and Green (2001). If individuals differ in terms of the effort they exert per hour, hourly wage is equal to effort per hour multiplied by productivity (which is then an _effort wage_ ). Returning to our previous example of the engineer and mechanic, it may be that in addition to working more hours per week than the mechanic, the engineer also exerts more effort per hour worked, and therefore the hours worked variation masks substantially more total labor supply variation (effort per hour multiplied by hours worked) between the two individuals. We now proceed to investigate what we can learn about income inequality from labor supply elasticities if individuals differ in terms of their effort per hour.

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# **4.1 Recovering Productivities and Preferences with Effort Decisions**

We consider the following generalized set-up, in which individuals choose both effort per<sup>15</sup> hour, _e_ , and hours worked, _h_ :<sup>14</sup> In this setup, which nests the model from Section 3, total labor supply is given by _he_ and is unobservable.<sup>16</sup> Unobservability of total labor supply complicates the analysis because productivity levels _n_ are no longer directly observable. Now we have _z_ = _nhe_ , so that hourly wages are equal to _ne_ , but because _e_ is not observable we cannot infer _n_ simply by observing hourly wage.<sup>17</sup> More generally, one can interpret this setup as allowing for two dimensions of labor supply, only one of which, _h_ , is observable to the economist. However, our model is easily extended to include even more components of labor supply so that income _z_ = _n_ ( _h_ 1 _e_ 1 + _h_ 2 _e_ 2 + _..._ + _hmem_ ). In this case, all we need to apply our method is to observe optimal income _z_<sup>_∗_</sup> and one component of labor supply _h_<sup>_∗_</sup> _i_<sup>(seeAppendixA.7).</sup> Our goal is unchanged: show how we can use labor supply elasticities to recover the function which maps optimal incomes and hours worked to productivities and preferences, _G_<sup>_−_1</sup> : _Z_<sup>_∗_</sup> _× H_<sup>_∗_</sup> _→ N × A_ . Even though we cannot directly observe individual effort _e_<sup>_∗_</sup> , we show that we are still be able to express (log( _n_ ) _,_ log( _α_ )) = _G_<sup>_−_1</sup> (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )) in terms of labor supply elasticities without any functional form assumptions. As in Section 3, the first step is understanding how differences in _n_ and _α_ manifest into differences in observables _z_<sup>_∗_</sup> and _h_<sup>_∗_</sup> :

> 14As before, we assume that the tax rate is constant. We show how the method can be applied with a piece-wise linear tax schedule with increasing marginal tax rates in Appendix A.4.

> 15Note, in our baseline model as well as this more general model, we assume that all individuals have the same level of unearned income. The methodology is also easily adapted to account for differences in unearned income if we can observe unearned income as well as the labor supply elasticities with respect to unearned income. This will allow us to subtract out the effect of unearned income on labor supply, which is entirely analogous to netting out gross substitution effects of different wage levels (see Appendix A.8).

> 16 If the cost of deviating from _e_ = 1 is infinite, then this model is equivalent to the model from Section

3.

> 17 This sort of model is discussed in, for example, Atkinson and Stiglitz (1976).

15
**Lemma 4.1.** _The derivative matrix of_ (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )) = _G_ (log( _n_ ) _,_ log( _α_ )) _is:_

_Proof._ See Appendix A.3.

We will no longer assume elasticities to be constant - they now vary with (log( _n_ ) _,_ log( _α_ )). The proof of Lemma 4.1 is just an application of the implicit function theorem and the intuition for Lemma 4.1 is very similar to the intuition for Lemmas 3.1 and 3.2 in the baseline framework without effort decisions. Changing _n_ leads to an income effect and a price effect, so leads to the same behavioral effect on hours worked as an uncompensated tax change; changing _α_ effectively changes the value of consumption, so leads to the same behavioral effect on hours worked as a compensated tax change. Additionally, Lemma 4.1 tells us how _incomes_ change with _n_ and _α_ , which is important because, unlike the baseline setup, _ξz_<sup>_u_=</sup><sup>_ξ_</sup> _h_<sup>_u_and</sup><sup>_ξ_</sup> _z_<sup>_c_=</sup><sup>_ξ_</sup> _h_<sup>_c_(asnowindividualscanadjusttheirlaborsupplyon</sup> both the effort and hours worked margins). Because log( _z_<sup>_∗_</sup> ) = log( _n_ ) + log( _h_<sup>_∗_</sup> ) + log( _e_<sup>_∗_</sup> ), changing _n_ affects _z_<sup>_∗_</sup> directly through a mechanical effect and indirectly through the behavioral effect _n_ has on optimal labor supply decisions ( _h_<sup>_∗_</sup> and _e_<sup>_∗_</sup> ). The behavioral effect of changing _n_ on _z_<sup>_∗_</sup> (i.e., the combined effect on _h_<sup>_∗_</sup> and _e_<sup>_∗_</sup> ) is equal to the uncompensated tax elasticity, hence the effect of _n_ on _z_<sup>_∗_</sup> is equal the mechanical effect plus the behavioral effect: _ξz_<sup>_n_= 1 +</sup><sup>_ξ_</sup> _z_<sup>_u_.Changing</sup><sup>_α_just leads to a behavioral response of income,</sup> which is again identical to the response from a compensated tax change, so _ξz_<sup>_α_=</sup><sup>_ξ_</sup> _z_<sup>_c_.This</sup> brings us to our main result: we can use observable labor supply elasticities to recover the function between incomes and hours and productivities and preferences: **Proposition 4.2.** _We can recover G_<sup>_−_1</sup> : _Z_<sup>_∗_</sup> _×H_<sup>_∗_</sup> _→ N ×A from the heterogeneous observable elasticities ξz_<sup>_u_(</sup><sup>_z∗, h∗_)</sup><sup>_,ξ_</sup> _h_<sup>_u_(</sup><sup>_z∗, h∗_)</sup><sup>_,ξ_</sup> _z_<sup>_c_(</sup><sup>_z∗, h∗_)</sup><sup>_andξ_</sup> _h_<sup>_c_(</sup><sup>_z∗, h∗_)</sup><sup>_aslongasallindividuals_</sup> ( _n, α_ ) _have elasticities satisfying ξz_<sup>_c>_0</sup><sup>_,ξ_</sup> _h_<sup>_c>_0</sup><sup>_,ξ_</sup> _h_<sup>_c≥ξ_</sup> _h_<sup>_u,andξ_</sup> _z_<sup>_u−ξ_</sup> _z_<sup>_c> −_1</sup><sup>_._18</sup> _Proof._ We prove Proposition 4.2 under the assumption of a _linear_ tax rate - the piece-wise linear case is slightly more complicated due to the presence of kink points. See Appendix A.4 for a derivation with a piece-wise linear tax rate with increasing marginal tax rates.

Let us define _G_ : _N × A → Z_<sup>_∗_</sup> _× H_<sup>_∗_</sup> as the continuously differentiable function that maps each (log( _n_ ) _,_ log( _α_ )) to a (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )).<sup>19</sup> Our goal is to find the inverse function, _G_<sup>_−_1</sup> : _Z_<sup>_∗_</sup> _×H_<sup>_∗_</sup> _→ N ×A_ . By Lemma 4.1, we can recover the Jacobian derivative

> 18Positive compensated elasticities are standard. Uncompensated elasticities being smaller than compensated elasticities (i.e., negative income effects) are also standard. Finally, the assumption _ξz_<sup>_u−ξ_</sup> _z_<sup>_c> −_1</sup> means income effects are not so extreme that individuals decrease income by more than $1 in response to a $1 increase in unearned income.

> 19 _G_ will be continuously differentiable as long as the utility function is twice continuously differentiable.

16
matrix of the function _G_ , denoted **J** _G_ :<sup>20</sup> We want to show that the mapping _G_ is a homeomorphism onto its image, i.e., that each ( _n, α_ ) chooses a unique optimal ( _z_<sup>_∗_</sup> _, h_<sup>_∗_</sup> ). In order to show that _G_ is homeomorphic, we need to first show that its Jacobian has an everywhere non-zero determinant, which is necessary for local invertibility. Dropping the arguments of the elasticities, the determinant of the Jacobian is: The first equality is an identity, the second is algebra, and the inequality comes from the assumptions that _ξz_<sup>_c>_0,</sup><sup>_ξ_</sup> _h_<sup>_c>_0,</sup><sup>_ξ_</sup> _h_<sup>_c≥ξ_</sup> _h_<sup>_u_,and</sup><sup>_ξ_</sup> _z_<sup>_u−ξ_</sup> _z_<sup>_c>−_1.Therefore,under</sup> the conditions stated in the proposition, **J** _G_ has a non-zero determinant. Moreover, (1 + _ξz_<sup>_u_)</sup><sup>_>_0(as(1 +</sup><sup>_ξ_</sup> _z_<sup>_u−ξ_</sup> _z_<sup>_c_)</sup><sup>_>_0and</sup><sup>_ξ_</sup> _z_<sup>_c>_0)and</sup><sup>_ξ_</sup> _h_<sup>_c>_0sothat</sup><sup>**J**</sup><sup>_G_haspositiveleading</sup> principle minors, hence is everywhere positive definite. A mapping _G_ on a convex domain with positive definite Jacobian matrix must be homeomorphic onto its image by Gale and Nikaido (1965) Theorem 6 (we assume the elasticity conditions hold for all ( _n, α_ ) _∈_ R<sup>2</sup> + so that the domain is convex).

Thus, the mapping _G_ is globally invertible; moreover, by the inverse function theorem, the Jacobian of the inverse mapping _G_<sup>_−_1</sup> is given by: From here, we simply pick a particular ( _z_ 0<sup>_∗, h∗_</sup> 0<sup>) and normalize (log(</sup><sup>_n_(</sup><sup>_z_</sup> 0<sup>_∗, h∗_</sup> 0<sup>))</sup><sup>_,_log(</sup><sup>_α_(</sup><sup>_z_</sup> 0<sup>_∗, h∗_</sup> 0<sup>))) =</sup> (0 _,_ 0). Finally, if _γ_ represents a path from (log( _z_ 0<sup>_∗_)</sup><sup>_,_log(</sup><sup>_h∗_</sup> 0<sup>))to(log(</sup><sup>_z∗_)</sup><sup>_,_log(</sup><sup>_h∗_)),wehave</sup> by Stokes’ Theorem:<sup>21</sup> Evaluating the path integral in Equation 6 allows us to match every optimal choice of income and hours, ( _z_<sup>_∗_</sup> _, h_<sup>_∗_</sup> ), to a unique level of ( _n, α_ ), i.e., to recover _G_<sup>_−_1</sup> . As an example,

> 20In practice, the observed Jacobian must additionally be consistent with some function _G_ , i.e., the Jacobian field must be conservative.

> 21We require that the set of observed ( _z∗, h∗_ ) values be path connected.

17
~~LJ~~ <!-- Start of picture text --> logz*<br>hy , log z3)<br>a<br>(log ho , log 29")<br>log h*<br><!-- End of picture text -->
from the identity _z_<sup>_∗_</sup> = _nh_<sup>_∗_</sup> _e_<sup>_∗_</sup> ) from his/her optimal choice of income and hours worked as well as the labor supply elasticities of income and hours worked with respect to the tax rate.

How can we use labor supply elasticities to infer optimal effort per hour from observable quantities _z_<sup>_∗_</sup> and _h_<sup>_∗_</sup> ? First, note that log( _e_<sup>_∗_</sup> (log( _n_ ) _,_ log( _α_ ))) = log( _z_<sup>_∗_</sup> (log( _n_ ) _,_ log( _α_ ))) _−_ log( _h_<sup>_∗_</sup> (log( _n_ ) _,_ log( _α_ ))) _−_ log( _n_ ). Under the conditions in Proposition 4.2, we can invert the relationship between ( _z_<sup>_∗_</sup> _, h_<sup>_∗_</sup> ) and ( _n, α_ ) so as to write _n_ and _α_ in terms of _z_<sup>_∗_</sup> and _h_<sup>_∗_</sup> . Hence, we can also write _e_<sup>_∗_</sup> as a function of _z_<sup>_∗_</sup> and _h_<sup>_∗_</sup> . We have that: For ease of explanation, let us assume that income effects are negligible. In this case we can show:<sup>23</sup> The first equality in Equation 8 comes from differentiating Equation 7 and the second _∂_ log( _n_ <u>)</u> equality uses the equation for _∂_ log( _h_<sup>_∗_</sup> )<sup>from the inverse Jacobian in Proposition 4.2.Next,</sup> note that _ξz_<sup>_c−ξ_</sup> _h_<sup>_c_isequaltotheelasticityofeffortperhourwithrespecttothetaxrate:</sup> Hence, Equation 8 is intuitive: individuals’ optimal effort per hour changes in proportion to their optimal hours worked in accordance with the ratio of the effort elasticity, _ξe_<sup>_c_=</sup> _∂_ log( _n_ <u>)</u> _ξz_<sup>_c−ξ_</sup> _h_<sup>_c_,tothehourselasticity,</sup><sup>_ξ_</sup> _h_<sup>_c_.Next,weuse</sup> _∂_ log( _z_<sup>_∗_</sup> )<sup>fromtheinverseJacobianin</sup> Proposition 4.2 to show that: Solving the system of partial differential equations given by 8 and 9 allows us to infer optimal effort for any optimal level of income and hours worked. If we again assume all relevant elasticities are constant, we have (for some constant _k_ , which we can normalize to 0 without loss of generality): Hence, by observing both the income and the hours elasticity (with respect to the tax rate), we can infer optimal effort decisions. Once we infer optimal log( _e_<sup>_∗_</sup> ) associated

> 23See Appendix A.9 for a discussion of where this formula comes from and how it changes with income

19
with each optimal level of (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )), we can recover log( _n_ ) = log( _z_<sup>_∗_</sup> ) _−_ log( _h_<sup>_∗_</sup> ) _−_ log( _e_<sup>_∗_</sup> (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> ))). Finally, we can recover _α_ from optimal hours worked by netting out the substitution effects of different effort wages _n_ as before using:<sup>24</sup> In summation, the intuition behind Proposition 4.2 has three steps. First, optimal effort per hour is related to optimal hours worked through the equation: log( _e_<sup>_∗_</sup> ) = _<u>ξz</u>_<sup>_c_</sup> _ξ_<sup>_−_</sup> _h_<sup>_<u>cξ</u>_</sup> _<u>h</u>_<sup>_c_</sup> log( _h_<sup>_∗_</sup> ); intuitively, optimal effort varies with optimal hours worked in the ratio of the effort elasticity w.r.t. the tax rate to the elasticity of hours worked w.r.t. the tax rate. Hence, we can infer optimal effort per hour from optimal hours worked. Second, once we know optimal effort per hour, we can recover productivity using _z_<sup>_∗_</sup> = _nh_<sup>_∗_</sup> _e_<sup>_∗_</sup> . Third, once we have recovered productivity, we can subtract out the component of hours worked due to productivity effects; the remaining component allows us to recover preferences.

To solidify ideas, recall our example of the engineer making $30/hour working 60 hours/week and the mechanic making $10/hour working 40 hours/week (assume both work 50 weeks/year). For purposes of illustration, suppose that the hours elasticity is half as large as the income elasticity so that the hours and effort elasticities are equal: _<u>ξe</u>_<sup>_c_Hence:</sup> _ξh_<sup>_<u>c</u>_= 1.</sup> Thus, _e_<sup>_∗_</sup> _eng_<sup>=</sup> 2<sup><u>3</u></sup><sup>_e_</sup> _mech_<sup>_∗_,i.e.theengineerexerts1.5timesasmucheffortperhourasthe</sup> mechanic. Normalizing _e_<sup>_∗_</sup> _mech_<sup>_≡_1andusingthefactthathourlywageisequalto</sup><sup>_ne∗_,</sup> we can deduce that the mechanic’s productivity is 10 and the engineer’s productivity is 20; hence, once we account for effort differences, we infer that the engineer is only twice as productive as the mechanic as opposed to three times as productive if we assume productivity is equal to hourly wage. We could then find _α_ for both individuals by netting log( _h_<sup>_∗_</sup> <u>)</u> _−ξh_<sup>_u_log(</sup><sup>_n_</sup><sup><u>)</u></sup> out substitution effects using log( _α_ ) = _ξh_<sup>_<u>c</u>_</sup> .

We have shown in this section that we can still use labor supply elasticities to infer individuals’ productivities and preferences even if individuals make unobservable effort decisions in addition to choosing how many hours to work. Importantly, there are four key elasticities we need to recover the inverse function used to infer productivities and

> 24While the formulas are slightly different, this intuition still goes through with heterogeneous elasticities. We can still solve the system of differential equations 8 and 9 to find optimal effort as a function of optimal labor supply decisions ( _z_<sup>_∗_</sup> _, h_<sup>_∗_</sup> ). We can still find log( _n_ ) = log( _z_<sup>_∗_</sup> ) _−_ log( _h_<sup>_∗_</sup> ) _−_ log( _e_<sup>_∗_</sup> ). Finally, we can solve differential equations 2 and 3 (replacing the function arguments as (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )), which is again without loss due to invertibility) to find log( _h_<sup>_∗_</sup> ) as a function of (log( _n_ ) _,_ log( _α_ )), and then invert this function to find log( _α_ ) as a function of log( _h_<sup>_∗_</sup> ) and log( _n_ ).

20
preferences: the uncompensated income and hours elasticities with respect to the tax rate and the compensated income and hours elasticities with respect to the tax rate ( _ξz_<sup>_u, ξ_</sup> _h_<sup>_u, ξ_</sup> _z_<sup>_c, ξ_</sup> _h_<sup>_c_);wewillinvestigatewhatthemagnitudesoftheseparametersimplyabout</sup> the sources of income inequality in Section 5.

But before we move on to investigate what empirical labor supply elasticity estimates imply about income inequality in the U.S., we discuss what we can learn from labor supply elasticities (i.e., how to interpret our findings) if productivity is partly determined by prior human capital acquisition (which may in turn have been due to differences in innate skills or preferences).

# **4.2 Dynamic Re-Interpretation**

All of our results have been derived in the context of a static labor supply model that abstracts from the possibility that individual productivities are partly due to past labor supply decisions or human capital acquisition. We show that even if individual productivities are driven by previous decisions, we can still use labor supply elasticities to recover individual preferences and productivities, recognizing that productivities are determined both by innate skills as well as past human capital acquisition. This is still an empirically interesting object as it tells us how much of income inequality is due to _cross-sectional_ productivities (at a given point in time) vs. preferences. Moreover, this yields a lower bound for the extent of income inequality due to preferences. This is because some of the cross sectional variation in productivities is in part due to differences in past decisions, which were in turn partially due to differences in preferences.<sup>25</sup> Consider a model in which individuals differ in terms of innate skills _n_ 0 and preferences _α_ and first make a human capital decision _K_ , at cost _κ_ ( _K_ ), and then for the rest of their life choose effort and hours worked each year conditional on this prior human capital decision. Furthermore, suppose that individuals’ effort wages grow endogenously over time. Let us denote this growth rate at time _t_ as _qt_ ( _ht, et_ ) and the cumulative growth _Qt ≡_<sup>�</sup><sup>_t_</sup> _s_<sup>_−_</sup> =1<sup>1</sup><sup>_qs_(</sup><sup>_hs, es_).Theindividualchoiceproblemcanbewrittenas:</sup> If we define _nt_ = _n_ 0 _KQt_ as the endogenous effort wage at time _t_ , then an analogue to Lemma 4.1 still holds as _ξz_<sup>_n_</sup> _t_<sup>_t_=1+</sup><sup>_ξ_</sup> _z_<sup>_u_</sup> _t_<sup>,</sup><sup>_ξ_</sup> _h_<sup>_n_</sup> _t_<sup>_t_=</sup><sup>_ξ_</sup> _h_<sup>_u_</sup> _t_<sup>,</sup><sup>_ξ_</sup> _z_<sup>_α_</sup> _t_<sup>=</sup><sup>_ξ_</sup> _z_<sup>_c_</sup> _t_<sup>,and</sup><sup>_ξ_</sup> _h_<sup>_α_</sup> _t_<sup>=</sup><sup>_ξ_</sup> _h_<sup>_c_</sup> _t_<sup>,see</sup> Appendix A.10. Hence, we can use Proposition 4.2 along with annual data on incomes

> 25Note, we assume preferences are constant over time; i.e., unlike productivities, we assume preferences are not affected by prior labor supply and human capital decisions.

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and hours worked (as well as the corresponding elasticities) to determine the function _G_<sup>_−_1</sup> : _Zt_<sup>_∗× H_</sup> _t_<sup>_∗→Nt× A_.Lastly,wecanextendthisideatoincludesavingsdecisions,</sup> see Appendix A.11.

Analyzing this dynamic setup is useful in so far as it clarifies the interpretation of our method: we use labor supply elasticities to learn about how much of income inequality is due to _cross-sectional_ productivities (at a given point in time) vs. preferences. Recognizing that this is the nature of the exercise, we now proceed to investigate what the empirically estimated labor supply elasticities imply about the drivers of income inequality in the context of the U.S.

# **5 Investigating What Labor Supply Elasticities Imply About Income Inequality in the U.S.**

In this section, we apply the methodology laid out in Sections 3 and 4 to data. In order to recover individual productivities and preferences, Proposition 4.2 tells us that we require labor supply elasticities of incomes and hours worked with respect to the tax rate. But the empirical literature has not reached a consensus on the magnitudes of these elasticities. Thus, the goal of this empirical exercise, which should be viewed primarily as a proof of concept, is to investigate what different labor supply elasticities imply about income inequality in the U.S.

Recall from Proposition 4.2 that there are four key labor supply elasticities (more precisely, elasticity functions) that underlie the inversion between labor supply decisions and primitives; these four elasticities are contained in the Jacobian matrix of partial derivatives of primitives with respect to labor supply decisions: While Proposition 4.2 allows for the these elasticities to vary across individuals with different incomes and/or hours worked, empirical estimation of labor supply elasticities has, for the most part, focused on recovering _average_ tax elasticities as opposed to heterogeneous tax elasticities. Hence, for our baseline estimates of elasticities, we will assume these elasticities are constant and use the average (compensated) income and hours elasticity estimates from a number of studies discussed in Chetty (2012): _ξz_<sup>_c_= 0</sup><sup>_._15 and</sup><sup>_ξ_</sup> _h_<sup>_c_= 0</sup><sup>_._15.26</sup> These parameter estimates correspond to an effort elasticity of 0 (as _ξz_<sup>_c−ξ_</sup> _h_<sup>_c_=0)and</sup> therefore can be interpreted in the context of Section 3, in which productivity is equivalent to hourly wage. Furthermore, consistent with most of the empirical literature on

> 26Chetty (2012) only discusses average compensated elasticities. Also, we assume that these elasticity values correspond only to real responses (as opposed to reporting responses).

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behavioral responses to taxation (e.g., Blundell and MaCurdy, 1999), we assume that income effects are negligible (so that _ξz_<sup>_u_=</sup><sup>_ξ_</sup> _z_<sup>_c_and</sup><sup>_ξ_</sup> _h_<sup>_u_=</sup><sup>_ξ_</sup> _h_<sup>_c_).27</sup> After briefly discussing what the baseline results imply about the drivers of income inequality, our main analysis concerns performing the inversion from labor supply decisions to primitives under different assumptions on the relevant elasticities, highlighting how deviations from the baseline estimates change our inference about the determinants of income inequality. We show that (1) larger differences between the income and hours elasticities with respect to the tax rate (i.e., larger effort elasticities) and (2) larger income effects will both lead us to infer that preference heterogeneity is increasingly important in driving income differences between rich and poor. We also discuss briefly at the end of this section how our findings change if allow for heterogeneity in the elasticity schedules (roughly in line with the findings of Gruber and Saez, 2002) and how we can account for labor market frictions using survey data on actual and preferred hours.

# **5.1 Data on Incomes and Hours Worked**

We will use data on incomes and hours worked from the American Time Use Survey (ATUS), which is a survey conducted on a subset of individuals who have participated in the CPS.<sup>28</sup> In addition to income data, the ATUS asks respondents to meticulously detail all of their activities on a particular (random) “diary day”. We then assume that this noisy “diary day” measure is representative of this individual’s average daily hours worked. We also do not have days worked per year, so we impute that all individuals work 250 days a year unless they report being part time and work _>_ 8 hours on their diary day, in which case we impute their days worked as 125. Our sample consists of all individuals reporting a positive income, thereby abstracting from the possibility of joint familial labor supply decisions. We show that our findings all hold with the smaller sample of single individuals, shown in Appendix C.3. We drop individuals who say they are involuntarily under-employed, hopefully mitigating the effect of labor supply frictions on our inferences. Our final sample from the ATUS then consists of data on (inflation adjusted) incomes and diary hours for 34,470 unique individuals from the years 2003-2015. See Appendix B for more detail on our sample construction.

Our measure of hours worked is noisy due to measurement and aggregation errors. Importantly, this noisy measure of hours worked is fine for our purposes so long as the noise is unbiased in the sense that the sample joint distribution of incomes and hours worked is representative of the true population joint distribution. Even if the sample

> 27We also assume that all individuals face a constant linear tax rate even though our method is easily adaptable to tax schedules with kinks, see Appendix A.4. This is for simplicity and consistency with the main body of the text and is likely inconsequential given the lack of bunching in the empirical income density.

> 28We discuss in Appendix B.2 why we do not use the hours worked measure from the CPS.

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distribution is not the same as the population distribution, we expect this should not affect the comparisons between the relative importance of productivities vs. preferences for different assumptions on the elasticity parameters.

# **5.2 Baseline Estimates**

Using our baseline estimates from Chetty (2012), _ξz_<sup>_c_=</sup><sup>_ξ_</sup> _z_<sup>_u_=</sup><sup>_ξ_</sup> _h_<sup>_c_=</sup><sup>_ξ_</sup> _h_<sup>_u_=0</sup><sup>_._15,wecan</sup> recover the function _G_<sup>_−_1</sup> (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )) using the inverse Jacobian from Proposition 4.2. Applying _G_<sup>_−_1</sup> (log( _z_<sup>_∗_</sup> ) _,_ log( _h_<sup>_∗_</sup> )) to the observed distribution of incomes and hours worked from the ATUS yields a value of ( _n, α_ ) for each individual in our sample. However, this distribution of productivities and preferences ( _n, α_ ) is not easily interpreted. Towards understanding the role productivities and preferences play in driving income inequality, we will construct the counter-factual income for each individual if (1) everyone had the same productivity or (2) everyone had the same preferences. Comparing these measures with actual income will help us understand the extent to which inequality is due to productivities vs. preferences.<sup>29</sup> First, for all individuals ( _n, α_ ) we will calculate _zn_<sup>_CF_</sup> 0<sup>=</sup><sup>_n_0</sup><sup>_h∗_(</sup><sup>_n_0</sup><sup>_, α_)</sup><sup>_e∗_(</sup><sup>_n_0</sup><sup>_, α_), the income</sup> they would optimally earn if they had productivity _n_ 0 and preferences _α_ . This is feasible because we have identified each person’s productivity _n_ and we know the manner in which both hours worked and effort per hour change with _n_ . Second, for all individuals ( _n, α_ ) we calculate _zα_<sup>_CF_</sup> 0<sup>=</sup><sup>_nh∗_(</sup><sup>_n, α_0)</sup><sup>_e∗_(</sup><sup>_n, α_0), the income they would earn if they had productivity</sup> _n_ and preferences _α_ 0; this exercise is possible because we have identified each individual’s preferences _α_ and we know how both hours and effort per hour change with _α_ .

In Figure 4a we plot average counter-factual incomes at each actual income level assuming all individuals had the same _n_ (the baseline level of _n_ 0 is chosen so that the mean income level in the counter-factual income distribution matches the mean income level in the empirical income distribution). In Figure 4b we plot average counter-factual incomes at each actual income level assuming all individuals had the the same _α_ (again, the baseline level of _α_ 0 is again chosen so that the mean income level in the counter-factual income distribution matches the mean income level in the empirical income distribution).

> 29While this counter-factual income exercise is nominally performed under the assumption that _α_ enters the utility function as _αu_ ( _c_ ) _− v_ ( _h, e_ ), all of the counter-factual income measures in this section are actually invariant to any functional form of preference heterogeneity for which income monotonically increases in preferences. Conceptually, as long as hours are increasing in preferences, our method recovers the correct preference rankings among all individuals (even if the nature of preference heterogeneity is wrong due to functional form mis-specification); we show in Appendix A.6 that the counter-factual incomes only depend on these ordinal preference rankings.

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4.2, we have only identified the sources of cross-sectional income inequality; hence, we have not ruled out the possibility that much of the observed cross-sectional heterogeneity in productivity is due to differences in past labor supply or human capital decisions. Such an investigation is beyond the scope of this paper, but we believe this is a useful area for further work.

# **5.4 Heterogeneity in Elasticities**

So far in this section we have assumed that all the relevant elasticity parameters are constant. We now consider how the results change if elasticities are heterogeneous. In particular, we consider two scenarios: (1) where elasticities linearly increase with log hours worked, and (2) where elasticities linearly decrease with log hours worked.<sup>31</sup> As in our baseline specification, we assume that income effects are negligible and that the effort elasticity is 0. The median elasticity is still _ξz_<sup>_c_=</sup><sup>_ξ_</sup> _h_<sup>_c_= 0</sup><sup>_._15,howeverwhenweallow</sup> the elasticity to increase with hours worked, the lowest hours-worked individuals have an elasticity of around 0 while the highest hours worked individuals have an elasticity of around 0.2. Conversely, when we allow the elasticity to decrease with hours worked, the lowest hours-worked individuals have an elasticity of around 0.6 while the highest hours worked individuals have an elasticity around 0.<sup>32</sup> We present the results with heterogeneous elasticities in Appendix C.1. In addition to illustrating how to implement our method when individuals have different elasticities, the main takeaway from this exercise is that the differences between our two scenarios with heterogeneous elasticities and our baseline scenario are very small. This is ultimately due to the fact that average hours worked are not changing substantially over the income space, implying that average elasticities are not changing substantially over the income space. Consequently, given our data on income and hours worked, the average elasticity is more important than differences in elasticities between high- and low-hours individuals for determining the relative importance of productivities vs. preferences.

# **5.5 Labor Supply Frictions**

We also consider how the presence of labor supply frictions affects our understanding about the determinants of income inequality. As discussed in Section 3.4, to make progress if there are labor market frictions, we need to know individuals’ optimal labor supply if

> 31Because individuals with higher hours worked also have higher incomes, allowing elasticities to increase with hours worked is consistent with the findings of Gruber and Saez (2002) who find that higher income individuals have higher elasticities. Conversely, one may expect elasticities to fall as hours worked rises reflecting the fact that there are only so many hours in a day. 32 _c_ With increasing elasticities we have _ξh_<sup>(</sup><sup>_h_)=0</sup><sup>_._15 + 0</sup><sup>_._05(log(</sup><sup>_h_)</sup><sup>_−_log(</sup><sup>_hmed_)),andwithdecreasing</sup> elasticities we have _ξh_<sup>_c_(</sup><sup>_h_) = 0</sup><sup>_._15</sup><sup>_−_0</sup><sup>_._15(log(</sup><sup>_h_)</sup><sup>_−_log(</sup><sup>_hmed_)),where</sup><sup>_hmed_ismedianhoursworked.These</sup> functions satisfy (a) _ξh_<sup>_c_(</sup><sup>_hmed_)=0</sup><sup>_._15and(b)min</sup><sup>_h∈H ξ_</sup> _h_<sup>_c_(</sup><sup>_h_)=0,where</sup><sup>_H_denotesthesetofobserved</sup> hours worked.

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they faced no frictions. We turn to the National Study of the Changing Workforce which, in addition to data on incomes and hours worked, contains data on preferred hours of work. We use this measure of preferred hours worked to recover productivities and preferences ( _n, α_ ) for each individual as discussed in Section 3.4. While labor market frictions are modest (optimal hours worked differ from observed hours worked by about 10% on average), they do not appear to be an overly large driver of income inequality relative to productivity and preference differences. We discuss our findings with frictions in more detail in Appendix C.2.

# **6 Application: Optimal Income Taxation**

In this section we analyze how labor supply elasticities impact the optimal extent of redistribution via the implied degree of income inequality due to heterogeneity in productivities vs. heterogeneity in preferences. We calculate optimal income tax schedules using the distribution of productivities and preferences recovered under the various assumptions on the magnitudes of labor supply elasticities in Section 5. We contrast these optimal schedules to the optimal schedules calculated assuming that all inequality is driven by productivity heterogeneity (as in Mirrlees, 1971 or Saez, 2001).<sup>33</sup> We find that (1) optimal tax rates are slightly higher than Mirrleesian optimal rates under our baseline elasticity estimates and (2) larger effort elasticities and larger income effects lead to lower optimal rates relative to the Mirrleesian case.

## **6.1 Optimal Tax Problem**

The optimal tax problem is to maximize social welfare, subject to a budget constraint and incentive compatibility constraints that individuals maximize utility conditional on the given tax schedule. Let us denote _c_<sup>_∗_</sup> ( _n, α_ ), _z_<sup>_∗_</sup> ( _n, α_ ) and _u_<sup>_∗_</sup> ( _n, α_ ) as the optimal consumption, income, and utility levels for individual ( _n, α_ ) under a given tax schedule. For some welfare weights _μ_ ( _n, α_ ), the government maximizes: The budget constraint is given by ( _E_ denotes government expenditures): 33 We assume the government can observe the distribution of incomes and hours worked so as to back out the distribution of productivites and preferences, but cannot condition the tax schedule on hours worked. If the government were to condition taxes on hours worked, then individuals would misreport their hours (as the government cannot feasibly monitor every person’s hours worked).

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The incentive compatibility constraints are that for all ( _n, α_ ), _z_<sup>_∗_</sup> ( _n, α_ ) is the optimal choice of income for type ( _n, α_ ) given the tax function.

Importantly, note that the optimal tax schedule can be vastly different depending on the distribution of _f_ ( _n, α_ ) if our tastes for redistribution (i.e., welfare weights _μ_ ( _n, α_ )) depend on the extent to which income levels are driven by _n_ vs. _α_ . Hence understanding the sources of income inequality, or _f_ ( _n, α_ ), is critical to performing welfare analysis.

# **6.2 Utility Functions**

For the purpose of an optimal tax simulation, we need to put a specific functional form on the utility function. For numerical simplicity, we consider two utility functions:<sup>34</sup> For utility function _U_<sup>(1)</sup> , the compensated (and uncompensated) elasticity is equal to _k_<sup><u>1</u></sup> (individuals with utility function _U_<sup>(1)</sup> have zero income effects). We choose _k_ to match the different elasticity estimates from Section 5. Moreover, as _c_ = _z − T_ ( _z_ ) = _neh − T_ ( _neh_ ), it is clear that agents only have disutility over total effort supplied, _eh_ . In other words, agents are indifferent between any combination of _e_ and _h_ that result in their optimal choice of _eh_ . We break this indifference by assuming that agents also have a constant hours elasticity equal to the observed hours elasticity. This technicality is not substantive - rather, it merely simplifies computations.

For our baseline labor supply elasticity estimates with the compensated elasticity equal <u>1</u> to 0.15 and zero income effects we use utility function _U_<sup>(1)</sup> and set _k_ = 0 _._ 15<sup>.Moreover,</sup> we assume that the hours elasticity is equal to 0.15 as well so that the effort elasticity is 0. For the labor supply elasticity assumption with a higher effort elasticity we also <u>1</u> use utility function _U_<sup>(1)</sup> and still have _k_ = 0 _._ 15<sup>,butwenowassumethattheeffortper</sup> hour elasticity is 0.1 instead of 0. For the labor supply elasticity assumption with large income effects, we have an uncompensated elasticity of 0 and a compensated elasticity of 0.15, so we use utility function _U_<sup>(2)</sup> , which has an uncompensated elasticity of 0 and a <u>1 1</u> compensated elasticity of 1+ _k_<sup>,so</sup><sup>_k_=</sup> 0 _._ 15<sup>_−_1.</sup>

> 34 Once we have specified a utility function, we can of course infer each individual’s ( _n, α_ ) directly from the first order conditions of each individual. Nonetheless, we believe the welfare exercise is useful to illustrate the importance of understanding the determinants of income inequality.

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# **6.3 Welfare Weights**

In order to conduct simulations, we must choose primitive welfare weights _μ_ ( _n, α_ ). Following Fleurbaey and Maniquet (2006) and Lockwood and Weinzierl (2016) we impose the normative criterion of preference neutrality, which mandates that redistribution is desirable when income inequality originates from productivity differences and undesirable if income inequality originates from preference differences; this framework is broadly consistent with the empirical/experimental relationship between beliefs over determinants of income inequality and redistributive tastes (e.g., Alesina et al., 2001 or Rey-Biel et al., 2011). More precisely, the welfare weights satisfy the criterion that if all income inequality is due to variation in preferences, the optimal tax schedule will be _T_ ( _z_ ) = 0 _∀z_ , which amounts to choosing _μ_ ( _n, α_ ) to equate marginal social utilities of consumption for all individuals with the same _n_ under _T_ ( _z_ ) = 0 _∀z_ . We impose that _μ_ ( _n,_ 1) = 1 _∀n_ , so that if all income inequality is driven by productivity differences, then the welfare function collapses to the un-weighted utilitarian welfare function as in Saez (2001).

We point out that, in general, simulating optimal tax schedules with multiple dimensions of heterogeneity is computationally difficult. In particular, Dodds (2019) shows that with multiple dimensions of heterogeneity, some individuals may not have a unique optimal income level under the optimal tax schedule (which causes so-called “jumping effects” when the tax schedule is perturbed), thereby rendering standard Hamiltonian optimization infeasible to calculate the optimal tax schedule. We avoid these complications by our choice of utility functions: as long as the distribution of productivities and preferences _f_ ( _n, α_ ) is continuous, and disutility of labor is convex ( _k ≥_ 0 in Equations 10 and 11), Proposition 7.4 in Dodds (2019) guarantees that all individuals will have unique optimal income levels, so that we can apply standard Hamiltonian optimization to solve the optimal tax problem with multiple dimensions of heterogeneity.

Computationally, we take a number of shortcuts which allow us to simplify the optimal tax problem. First, we calculate the set of ( _n, α_ ) who locate at each income level - <u>1</u> individuals with the same value of _v_ = _nα_ 1+ _k_ all choose the same income. We can refer to _v_ as the unified-type (following Lockwood and Weinzierl, 2016). Then we use our density of productivities and preferences _f_ ( _n, α_ ) to calculate the density of individuals with each unified type _v_ . Moreover, we can calculate the average welfare weight for each unified type _v_ using _f_ ( _n, α_ ) as well as _μ_ ( _n, α_ ). Then, once we know the density and average welfare weight at each unified type _v_ , we can simply apply the standard one dimensional Hamiltonian optimization approach as in Mirrlees (1971).<sup>35</sup>

> 35We explain the simulation procedure in more detail in Appendix D.

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# **6.4 Simulation Results**

We present optimal tax schedules using the distributions of productivities and preferences from Section 5 that correspond to (1) the baseline labor supply elasticity estimates from Chetty (2012) ( _ξz_<sup>_c_=</sup><sup>_ξ_</sup> _h_<sup>_c_=</sup><sup>_ξ_</sup> _z_<sup>_u_=</sup><sup>_ξ_</sup> _h_<sup>_u_=0</sup><sup>_._15),(2)alargereffortelasticity(</sup><sup>_ξ_</sup> _z_<sup>_c_=</sup><sup>_ξ_</sup> _z_<sup>_u_=</sup> 0 _._ 15 _, ξh_<sup>_c_=</sup><sup>_ξ_</sup> _h_<sup>_u_= 0</sup><sup>_._05),and(3)largerincomeeffects(</sup><sup>_ξ_</sup> _z_<sup>_c_=</sup><sup>_ξ_</sup> _h_<sup>_c_= 0</sup><sup>_._15</sup><sup>_, ξ_</sup> _z_<sup>_u_=</sup><sup>_ξ_</sup> _h_<sup>_u_= 0).Along</sup> with each optimal tax schedule we plot the Mirrleesian optimal schedule that assumes all income inequality is due to productivity differences. The optimal tax schedules corresponding to the baseline case, larger effort elasticity case, and larger income effects are shown in Figures 10a, 10b, and 10c, respectively. We choose to plot _average_ tax rates (as opposed to marginal tax rates, which can be found in Figure 23 in the Appendix) as this conveys the tax burden at each income level under the different distributions of productivities and preferences implied by the different values of the labor supply elasticities.<sup>36</sup> Note that different elasticity estimates imply different efficiency costs of taxation (so that the Mirrleesian benchmark is not constant across all the different elasticity estimates). The important aspect to focus on then is the _difference_ in tax rates between the Mirrleesian benchmark and our optimal tax schedules that account for both productivity and preference heterogeneity driving income inequality; this difference in tax rates is not driven by differences in the efficiency costs of taxation but by differences in the equity of taxation.

In Figure 10a, optimal average tax rates computed assuming both _n_ and _α_ heterogeneity are relatively similar to, but (almost) everywhere _≈_ 2 p.p. _higher_ than, the benchmark Mirrleesian rates, which assume all income inequality is driven by _n_ heterogeneity. This is because, under our baseline elasticity estimates, productivity heterogeneity drives most of income inequality and high income individuals have lower preferences on average, so that redistributing away from them is slightly more desirable than in the Mirrleesian benchmark. On the other hand, in Figures 10b and 10c (which correspond to a higher effort elasticity and larger income effects, respectively) we find that average tax rates are now lower than the Mirrleesian benchmark. Higher effort elasticities and larger income effects both lead to lower optimal tax rates relative to the Mirrleesian optimal tax schedule. This is because higher effort elasticities and larger income effects both imply that high income individuals have higher preferences for consumption, so that redistributing away from them is less desirable than in the Mirrleesian benchmark.

> 36Note, all individuals receive a lump-sum transfer under every optimal schedule. This transfer is increasing with overall tax rates and is excluded from income when calculating average tax rates.

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<!-- Start of picture text --> 08<br>— — Both nand a heterogeneity<br>07 ——— Only n heterogeneity<br>06 ~--- 2 =<br>2 Ss<br>& 05 =<br>x g<br>fi<br>© 04<br>D£<br>2 03<br><x<br>0.2<br>0.1<br>0<br>0 50000 100000 150000 200000<br>Annual Income<br><!-- End of picture text --> <!-- Start of picture text --> 08<br>— = Bothnand a heterogeneity<br>07 ——— Only n heterogeneity<br>06<br>2<br>B05<br>> 04<br>£<br>$03<br>¢<br>02<br>0.1<br>050000 100000 150000 200000<br>Annual Income<br><!-- End of picture text --> <!-- Start of picture text --> 08<br>— = Both nand a heterogeneity<br>07 ——— Only n heterogeneity<br>06 _--<--=<br>2 Teert ---<br>gO -7<br>& ‘<br>@ 0-4 7<br>£ i"<br>So3/ |<br>¢<br>02<br>0.1<br>0 50000 100000 150000 200000<br>Annual Income<br><!-- End of picture text -->
effects therefore imply lower tax rates relative to a Mirrleesian benchmark. The overall takeaway then is that tax elasticities are important not only for understanding efficiency costs of taxation, but also for understanding the equity benefits of taxation.

Finally, under all of the elasticity estimates considered, productivity heterogeneity is far more important in driving income inequality than is preference heterogeneity. However, our measure of hours worked is measured with some degree of error so that this result should be taken with a grain of salt; implementation of our methodology could be performed far better with a purpose-built dataset designed to more accurately measure hours worked. Moreover, interpreted in the context of a dynamic model as in Section 4.2, we have only identified sources of cross-sectional income inequality; hence, we have not ruled out the possibility that much of the observed cross-sectional heterogeneity in productivity is due to differences in human capital acquisition, which is in turn due partly to differences in preferences. As such, investigating the extent to which cross-sectional productivity differences are due to innate skills differences vs. human capital acquisition is a useful direction for further research.

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