Job Amenities & Earnings Inequality
Alex Bell∗
March 21, 2024

Abstract
Previous research on prices of job amenities has suffered from simultaneity bias
due to workers’ unobserved offer sets, resulting in “wrong-signed” compensating wage
differentials. I propose a simple amenity pricing framework that uses an imprecise
proxy for workers’ offer sets to identify amenity prices holding offer sets fixed. Using
price estimates for a set of observed job characteristics across various public survey
datasets, I find a large role for costly amenity substitution in explaining the gender
pay gap (on the order of two-thirds) and little role for amenities to explain inequalities
by race or by parent background.

JEL Classifications: C18, C39, C51, J31
Keywords: compensating differentials; inequality; occupational choice; gender; race
∗ University of California - Los Angeles. Contact: alexbell@ucla.edu. I am grateful to Raj Chetty,
Roland Fryer, Nathaniel Hendren, Lawrence Katz, and Elie Tamer for research advice during graduate
school. This work has also benefited from insightful conversations with Isaiah Andrews, Gary Chamberlain,
Kirsten Clinton, David Cutler, Tanaya Devi, Natalia Emanuel, Edward Glaeser, Claudia Goldin, Sandy
Jencks, Corinne Low, Philip Marx, Elizabeth Mishkin, Corina Mommaerts, Christian Moser, Alex Olssen,
Amanda Pallais, Ariel Pakes, Jonathan Roth, and Yulya Truskinovsky. Finally, I am deeply grateful for theinsights and encouragement of Sophie Calder-Wang and Shusheng Zhong, who have become my coauthors
on a companion paper advancing similar ideas in the housing context. This research has been supported by
Harvard’s Multidisciplinary Program in Inequality & Social Policy, the EO Foundation, and the Washington
Center for Equitable Growth. The views expressed herein are my own and do not necessarily reflect theviews of these organizations.

1

I

Introduction

Workers differ both in terms of total compensation and how they split their compensation
into wages and amenities. Determinants of total compensation may include productivity,
discrimination by employers, search frictions, and rents. But income differences also arise
when workers trade off between wages and other desirable attributes. This paper puts
forward a new methodology for estimating the trade-offs that workers face between income
and observable non-monetary characteristics of their jobs. These price estimates allow me
to discern whether workers lie on the same or different frontiers of total compensation. I
decompose earned income gaps by gender, race, and parent background into a component
that is due to sitting on different wage-amenity frontiers (differences in total compensation)versus differences along the wage-amenity frontiers (differences in valuations).
The idea that job amenities command a wage premium, or compensating wage differential, dates back at least to Smith (1776). Two centuries, later Brown (1980) best summarized
the existing empirical literature as holding “some clear support for the theory but an uncomfortable number of exceptions.” More recently, quasi-experimental evidence has revived
interest in the theory. A persistent barrier to studying the full impact of compensating
differentials on income inequality is the inherent difficulty of estimating prices for a wide
set of job amenities in a sample representative of the labor market.
The problem stems from the researcher’s inability to isolate workers who had similaroffer sets to each other but chose to split their compensation differently into wage andamenities. A classic illustration is that of physical safety on the job. Supposing workers
tend to prefer jobs with higher levels of physical safety, one would expect workers in safer
jobs to be paid less. Figure 1 shows the opposite is true in the NLSY. In general, safer
jobs pay more, not less. The goal, however, is to look at how income and safety vary
among workers who share the same level of productivity, and therefore could have obtained
similar types of jobs. Panel B shows that although controlling for observable productivityrelevant characteristics weakens the relationship, in this case it does not even change itto the expected sign. Such a result might be expected if the productivity channels that
determine a worker’s compensation are not fully observed. In the absence of a full set ofcontrols, income and safety are both determined by differences in (unobserved) productivity
and preferences. The regression of wage on safety conflates compensating differentials withthe returns to skill, much in the same way that a regression of quantities on prices conflates
supply and demand. Controlling for a few observed ability measures has done as muchto solve this simultaneity problem as controlling for a few shifters of supply would in aregression of price on quantity.1
1 Another analogy can be drawn to discrete choice analysis, in which choices are regressed on productcharacteristics (typically using logistic regression). That framework assumes that binary choices are informative of which options are better. My framework is similar, but uses an observed measure of worker
productivity rather than choice. The typical estimating equation for compensating differentials in represen-2Figure 1: The Classic Problem-.50Income SD's
.5

1

1.5

(a) Income by Safety

-3

-2

-1
Safety SD's

0

1

Slope = .1506 ( .0345)-.50Income SD's
.5

1

1.5

(b) With Ability Controls

-3

-2

-1
Safety SD's

0

1

Slope = .0235 ( .0254)

Notes: Panel A is a binned scatterplot of income on ten bins of safety. The red line is thebest-fit line on the unbinned data. Its slope and standard error are noted on the bottom
left of the graph. Panel B includes controls that attempt to proxy for worker ability: yearsof education, age, and AFQT. Data are NLSY respondents in 2012 and standard errors are
clustered at the occupation level; additional information on data is contained in Section
V.A.

3

The core contribution of this paper is to demonstrate the data used to construct Figure
1 is more than sufficient to remove the simultaneity bias previously seen as inherent toobservational analysis of compensating differentials. Similar to instrumental variables, myproxy approach can be viewed as treating an observed measure of ability as a shifter ofthe offer set.2 It differs from IV estimation in that the variable that needs to be shifted –the offer set – is not observed.3 Rather than “proxy,” some researchers also refer to this as
the “anti-instrument” approach, where the prefix “anti-” denotes opposite,4 referencing thefact that the opposite variable (i.e., the residual) is being instrumented from what we would
traditionally think of as IV. This approach also shares some similarity with the selection-onobservables literature (Altonji et al., 2005; Oster, 2019), which is described in more detailin Section I.A.
The OLS approach to tracing out wage-amenity variation generated by preferences requires the researcher to control for the full set of variables that determine workers’ abilities.
In contrast, my approach requires only one such variable. Both approaches, it should be
noted, are sensitive to a normalization on the part of the researcher as to which variable(or variables) belong as proxies for the offer set, rather than correlates of preferences. My
approach allows me to explore the robustness of results when the offer set is proxied by any
one of various types of cognitive, social, and physical characteristics of workers.
Figure 2 illustrates the difference between OLS and my proposed strategy using hypothetical data constructed to mirror the important facets of real labor market data. Suppose
the researcher observes only three coarse bins of workers’ ability, represented by colors. The
correspondence between observed ability and the actual offer set is quite noisy; some observably low-skilled workers enjoy very high wage and amenity. Panel A depicts the standard
regression of wage on amenity with indicators for observed ability (as an average of regressions within observed ability levels). The estimating equation is as follows, with coefficientof interest β̂:5Eˆ∗ [wage|amenity, obs. ability] = β̂amenity + I[obs. ability]
Because I observe workers’ true abilities with noise,6 within every level of observed ability
tative datasets is analogous to a regression of one product characteristic on other characteristics controllingfor which product was chosen, which is generally not consistent as described by McFadden (1973).
2 Other factors determining the offer set may include unobserved ability, but also factors typically viewed
as “stochastic” such as search frictions, connections, or discrimination.
3 My approach shares this feature in common with Olley and Pakes (1996), who describe how observed
investment can identify the unobserved productivity term in production functions, as well as Combes et al.
(2021) in the context of latent quality in the housing production function.
4 (e.g., anti-protons spin in the opposite direction as protons.)
5 I use notation Eˆ∗ to denote regressions throughout the paper (the linear conditional expectation estimator).
6 The ability bias problem can be framed either as measurement error (observed variables like education
are effectively noisy measures of the offer set) or analogously as omitted variables (some relevant controls
are not observed).

4

Figure 2: Graphical Intuition(a) OLS(b) Proposed Estimator

Notes: This figure describes two estimation strategies on hypothetical data in which higherability workers get more wage and amenity, such that a trade-off exists among workers ofa given ability. True ability is imperfectly observed by the researcher in only 3 rough bins,
which I call “observed ability.” The dots represent workers and are shaded in accordance withobserved skill level, with blue representing lowest, then red, then green. Panel A describes
a regression of wage on amenity with fixed effects for 3 levels of observed ability. The
fixed effects estimator here is depicted as the within-ability estimator of wage on amenity,
and the final average over all ability levels is given by the black line. The upward-sloping
OLS estimate mirrors the finding of Figure 1. Panel B describes the proposed estimatorof observed ability on wage and amenity, described in more detail in the text. The slopeof the regression line labeled Step 1, which tells us the direction in which observed ability
increases, is π̂δ̂ . Of greater interest, the line orthogonal to that regression line, labeled Step2, corresponds to wage-amenity variation when skill is held constant. The formula for theslope of that line is the negative inverse of the regression line, − π̂δ̂ , which corresponds tothe true price of the amenity in this model. Alternatively, the exact estimate could havebeen found by obtaining predicted values from the regression of observed ability on wage
ˆand amenity (denoted ability),
then regressing wage on amenity with those predicted valuesˆ
as a control. Approximate levels of ability
appear on the graph as colored bandwidths.

5there is still an upward-sloping relationship between wage and amenity. This explains theprevious upward-sloping relationship of Figure 1, even with controls for observed ability.
In contrast, Panel B sketches two ways to implement my proposed estimation strategy
in a linear framework. Both approaches begin the same way. Rather than using observed
ability as a control, it is the outcome of a regression on both wage and amenity. I call thisStep 1 on the graph:Eˆ∗ [obs. ability|wage, amenity] = δ̂wage + π̂amenity
Although the regression of Step 1 is not structural, it does provide some intuition. If wage
and amenity are two forms of compensation that workers enjoy, I would expect both δ̂ andπ̂ to be positive. The relative magnitudes of the coefficients reveal the tendency for higherability workers to get more wage and more amenity. The regression also has a graphicalinterpretation in wage-amenity space. The line points in the direction in which observed
skills – and thus by assumption offer sets – are increasing. This direction is orthogonal towhat I want to estimate; I seek to estimate the wage-amenity trade-off when the offer set
is held constant. The orthogonal direction is labeled as Step 2 in the graph, and it canbe obtained either from a ratio of coefficients or using predicted values from Step 1. Ina linear model, the two approaches produce numerically equivalent estimates (much like
instrumental variables and control functions).
The simplest approach is to read off the perpendicular line to the regression line of Step 1.
If Step 1 told us the direction in which skill increases, the orthogonal direction must describe
wage-amenity variation that holds skill constant, which is the relative price a worker faces.
Because the regression line of Step 1 has a slope of π̂δ̂ , the slope of the orthogonal line andthus the price estimate is − π̂δ̂ . I refer to this procedure as the ratio-of-coefficients approach.
An alternative approach uses predicted values from the regression of Step 1. The strengthof the predicted-values approach, as I elaborate on in Section III, is that it can easily accommodate a non-parametric estimator in place of the linear regression in Step 1. Referringˆto these predicted values as ability,
Step 2 of the predicted-values approach is to regress
wage on amenity with the inclusion of the constructed control:ˆˆEˆ∗ [wage|amenity, ability]= β̂amenity + γ̂ ability
This regression also need not be linear, although β̂ from the linear regression is numericallyequivalent to the ratio of coefficients, − π̂δ̂ .7 The predicted values of the first stage serve as an
ideal control for the offer sets. If the ability proxy was a noisy measure of the offer set, thenthat noise is purged in taking the predicted values. Critically, in order to correctly write the
7 In this case, because abilityˆ
is a linear combination of the outcome and a regressor, this approach yields
a perfect fit; standard errors from the second-stage regression are not meaningful (I discuss identificationrobust inference in Section IV.B).

6

offer set as a function of wages and amenities, it must be the case that there is only one latentvariable in the estimating equation that relates workers’ wages to amenities. I refer to thislatent variable interchangeably as the level of the offer set or the total compensation levelof the worker. It is this sole latent variable that will be proxied by observed qualifications.8
Whereas either estimation approach overcomes the problem of ability bias in observational data, a key limitation remains: in observational data, I still cannot price characteristics that I cannot observe. Real workers make choices taking into account many aspectsof jobs. Although my estimator readily extends to pricing many amenities jointly, I do notclaim to observe the full set. As I discuss in Section V.D, the likely “bundling” together ofobserved and unobserved amenities would cause my price estimates to be larger than thosethat priced the full set of amenities.9 Though the problem of unobserved amenities calls
for caution when interpreting price estimates, a positive correlation between observed andunobserved amenities if anything would strengthen my analysis of demographic pay gaps.
The fact that I cannot price all the dimensions in which work differs suggests that I am
likely to under-state the importance of amenities as a whole in pay gaps. But to the extent that many of the good amenities are bundled, an analysis that only takes into accountthe observed ones – at prices inclusive of how unobserved ones covary – may be a morereasonable approximation of the overall role of amenities.
Empirically, this paper’s main findings on demographic pay gaps seem unlikely to be
driven by problems of unobserved amenities. For instance, I find that conditional on total
wage and amenity compensation, women tend to earn better amenities and lower income
than men. This finding implies amenities play a substantial role in the current gender pay
gap, which seems difficult to explain by the problem of observing too few amenities. In stark
contrast, I find more limited roles for the same set of amenities to explain gaps by race andby parent background. Unlike the gender gap, these income inequalities are shown to be
primarily absolute gaps in total compensation rather than a function of how workers split
their compensation into wages and amenities. Although the possibility still remains that
8 To build further intuition for this predicted-values approach, recall that the basic structure of the
problem is that workers’ wages and amenities are both caused by differences in offer sets and preferences.
The goal is to build a control variable for offer sets such that at a given level of the control, workers stillobtain different combinations of wages and amenities (due to preferences). Thus, the ideal control variable
for the offer set can be written as some combination of wages and amenities. A prediction exercise ofobserved ability (in this case, a linear regression) guides the construction of that ideal control variable. In alarge enough sample, I would obtain the same predicted values if either true offer sets or a noisy proxy forthe level of the offer set were used as the dependent variable in Step 1.
9 One operationalization of a compensating differential, to which I do not adhere in this paper, is thewage trade-off a worker faces when an amenity increases by a unit, and all other job attributes are held
constant. Instead, I estimate the trade-offs workers face when they choose a typical job that has a unit
more of the amenity. Empirically, I find that observed amenities are correlated with each other: their prices
tend to be smaller when more amenities are priced together. If my observed amenities can be considereda random sample of all amenities, it seems likely that such a positive correlation would also exist between
observed and unobserved amenities (Altonji et al., 2005; Oster, 2019). This general bundling of amenities islikely to lead my price estimates to be larger in absolute magnitude than price estimates that hold all otheramenities fixed.

7additional racial and intergenerational income inequality is generated by amenities otherthan those I observe, my findings are consistent with a literature that has focused on therole of the absolute factors of skills and discrimination in causing these gaps. My findingson the gender gap are also consistent with a modern literature pointing to the importanceof how work is remunerated.
The outline for the rest of the paper is as follows. In the remainder of this section, I relate
my work to previous research on compensating differentials, including previous attempts todiagnose and solve the problem of ability bias highlighted by Figure 1. In Section II, I
provide my rendition of the usual structural model of compensating differentials; I discuss
the identification assumption in the context of that structural model and in the reducedform. Section III proves formally that the Rosen frontier is non-parametrically identified
by observational data, and Section IV discusses estimation in the linear model, including
potential sources of bias. Section V explores amenity price estimates in the NLSY linked with
O*NET occupation data. Section VI relates these price estimates to differences in quantitiesof amenities by gender, race, and parent background. Finally, Section VII concludes. Thisestimation strategy is portable to other topics in economics, and a companion paper applies
a similar strategy to pricing residential amenities (Bell et al., 2023).

I.A

Relation to Literature Pricing Job Amenities

This paper builds closely on the conceptual framework of a labor market sorting equilibrium
put forward by Thaler and Rosen (1976) and Rosen (1986). The only stylistic differenceis that I more explicitly model unobserved skill heterogeneity. The potential bias fromunobserved skill heterogeneity in such a model has been shown and calibrated by Hwang etal. (1992) (whose final paragraph nicely motivates this paper):
Where then should future research on the estimation of compensating wage
differentials be directed? In our opinion, significant progress is most likely tocome from new econometric methods that are better able to address the problem
of unobserved productivity heterogeneity. We hope that this study will stimulate
renewed efforts along this line.
An early wave of empirical literature on compensating differentials attempted to find evidence for the theory in nationally representative datasets, and the present paper uses manyof the same datasets. Lucas (1977) offered “some of the first systematic empirical support
for Adam Smith’s notion of equalizing wage differentials.” Lucas linked individual-level datafrom the Survey of Economic Opportunity with newly available occupational-level data fromthe Dictionary of Occupational Titles.10 Six categories of occupational characteristics were
investigated, including sedentariness, repetitiveness, and physical conditions such as risk of
10 The SEO is now part of the PSID and DOT is now O*NET.

8

injury. These job characteristics were priced in hedonic hourly wage equations, with productivity differences handled by controls for race, gender, age, years of education, and union
membership. Results were imprecise and mixed, but generally indicated that workers taking
on repetitive work and work in poor physical conditions – both thought to be undesirable –were earning slightly higher wages as a result.
Brown (1980) suspected that previous studies testing relationships between wage andamenities controlling for observed productivity measures were biased by “the omission of important worker abilities.” To better control for unobserved productivity differences acrossworkers, Brown introduced worker fixed effects into a panel derived from the NLSY also
linked with DOT occupation data. Ultimately, Brown’s within-worker comparisons of wages
and amenities proved quite similar to the cross-worker comparisons – that is, often apparently wrong-signed.11 My interpretation of this result is that even within-worker, there is agreat deal of idiosyncrasy in total realized utility across job spells. Rather than differencesin jobs across workers being caused by worker-level skill, this result suggests a great role forthe type of idiosyncrasy in overall utilities emphasized by Mortensen (2003), which I model
explicitly in the next section.
Sorkin (2018) also leverages worker moves, but across firms rather than across occupations, and finds some evidence in support of compensating differentials. Under assumptions
about what information is conveyed when workers move across firms, Sorkin (2018) estimates that 15% of the overall variation in earnings across the economy is due to firm-levelcompensating differentials. A key identifying assumption of this method is that preferences
are approximately stable over workers’ lifetimes, such that the central tendency of moves
represents better jobs, as opposed to jobs with particular amenities.12 While the identifying
assumption of this work contrasts with that of the present paper, the premise that verticaland horizontal differentiation exists in the labor market–with the vertical differentiation
confounding the estimation of compensating differentials–is a strong point of agreement.
Moreover, the methods are far more complimentary than they are in conflict with eachother; for instance, Bell et al. (2023) shows how a method similar to that of Sorkin (2018)
can be combined with the proxy approach to yield more credible estimates of amenity prices
than simply using the rank estimates as a control.
As is the case in all fields of economics, additional assumptions can yield identificationof additional parameters. Like this paper, Lamadon et al. (2022) also put forward a modelof the labor market that includes both horizontal and vertical differentiation of workers andfirms. While their model allows them to estimate not only compensating differentials, butalso imperfect competition and rent sharing, this comes at the cost of stringent functional
11 Villanueva (2007) also follows a within-worker approach in Germany to examine price heavy workload,
job insecurity, poor hours regulation, and skill mismatch. The author concludes the results “broadly support
the existence of” compensating differentials in Germany.
12 Subsequent work has leveraged this methodology both in labor applications (Sorkin, 2017; Morchio andMoser, 2020; Lehmann, 2023) and in housing (Fogel, 2021; Bell et al., 2023).

9form assumptions that are not present in this paper (for instance on the form of preference
heterogeneity). Taber and Vejlin (2020) also make a number of distributional and functional
form assumptions that allow them to estimate separately the forces of compensating differentials, search frictions, and skill heterogeneity upon labor market entry. While models thatunify several key sources of wage dispersion are critical to advancing our understanding ofincome inequality, the goal of the present paper is to establish a minimal set of assumptions under which researchers can estimate compensating differentials from observational
datasets, and these assumptions do not entail parametric distributions or functional forms.
In a different vein, much recent literature has used quasi-experimental methods to isolate
variation in wage due to particular job characteristics. For instance, by combining longitudinal survey data with seasonal variation in safety conditions in the Bering Sea, Lavetti(2018) was able to illustrate that commercial fishermen in this area are paid higher wages
to take on more fatality risk, and that this premium is falling as risk levels rise. Other
quasi-experimental work to estimate the value of statistical life have included the use of
mandated speed limits by Ashenfelter and Greenstone (2004) and military re-enlistment by
Greenstone et al. (2015).
The key advantage of quasi-experiments, relative to the present paper, is that they canrecover price estimates that control for unobserved amenities. This suggests an opportunity
to check my findings for bias due to omitted amenities. However, the vast majority ofthe quasi-experimental compensating differentials research has studied only the price ofamenities relating to safety and fatality risk, and typically in various sub-populations thatare unrepresentative of the labor market. The findings of this literature seem to be thatsafety is positively valued by the labor market; my findings certainly agree on this amenity.
There is not enough variation in fatality risk in my occupation-level data to more precisely
line-up with quasi-experimental estimates of the value of statistical life (although my point
estimate is in the same direction).
A related but different literature has focused on estimating workers’ valuations of amenities, rather than market prices.13 To clarify this distinction, a group of workers may value
an amenity a great deal, but this fact alone does not imply they sacrifice wages to get it;that depends on the price. Rather, a price is an equilibrium between workers’ heterogeneous
valuations of an amenity and firms’ heterogeneous costs of providing it. The typical strategy
to study workers’ valuations of amenities has been to survey particular groups of workers
about real (or fictitious) offers, and observe which offers they do (or would) accept. For
instance, a recent survey of undergraduates by Wiswall and Zafar (2018) found that female
students tended to state a higher willingness to pay for job flexibility and stability, whereas
male students more often stated preferences for earnings growth.14 A more representative
13 There is a similarity between my estimating procedure and the estimation procedure used by studies

that observe real or hypothetical worker choice data; this similarity is discussed in Online Appendix J.
14 Wiswall and Zafar (2018) collected preference data on six categories besides earnings: (1) earnings

10

survey by Maestas et al. (2018) also found large differences by gender in willingness to pay
for amenities, particularly for physical activity and paid time off.15 In a different vein, a pioneering survey of postdoctoral biologists by Stern (2004) focused on scientists’ valuation ofresearch freedom. Despite little relationship in the cross section between wage and freedom
to publish – due to ability bias16 – biologists’ job choices tended to be consistent with apositive valuation of research freedom. Mas and Pallais (2017) also applied a stated-choiceframework regarding job flexibility to real applicants at a call center and in a nationally
representative internet survey, and found slightly higher valuations of worker-friendly work
arrangements among women than men.
The empirical approach of this paper was heavily influenced by a much earlier literatureevaluating the potential for proxy approaches to overcome bias from unobserved ability. A
concise summary and relation to that literature is contained in Appendix K.
Finally, the assumption that a single imperfectly observed variable exists in the neoclassical amenity pricing equation connects deeply to a growing literature on selection corrections,in particular as put forward by Altonji et al. (2005) and Oster (2019). In the presentation
by Oster (2019), the estimator that I propose is numerically equivalent to a selection correction that sets RM ax = 1, which was also the assumption of Altonji et al. (2005).17 While
this parallel is important to bear in mind, the purpose and application of the two methods
contrast with each other. Whereas the selection-on-observables literature aims to estimate
an endogenous treatment effect, the method developed in this paper is tailored around estimating an implicit price in the face of a well documented simultaneity problem. Alongthese lines, a helpful analogy to characterize the relationship to the existing selection-onobservables literature lies in traditional IV estimation. In some cases, instruments are used
to estimate treatment effect parameters. But in other cases, instruments are used to resolve
the type of structural simultaneity that is thought to be caused by supply and demand;
this is where the present paper fits into the analogy. To my knowledge, there is no existing
work that adapts the selection-on-observables literature to topics characterized by structural
simultaneity like the questions of job choice that this paper studies.
growth, (2) number of hours, (3) part-time option (“flexibility”), (4) likelihood of being fired (“stability”),(5) performance compensation, and (6) proportion men.
15 Maestas et al. (2018) collected preference data on schedule flexibility, telecommuting opportunities,
physical job demands, pace of work, autonomy, paid time off, teamwork, opportunities for job training, andmeaningfulness.
16 Stern (2004) termed this the “productivity effect.”17 In the context of compensating differentials, a sufficient (but not necessary) assumption to justify
RM ax = 1 is if the vertical dimension of worker ability is the only latent variable the researcher believes
exists in relating wages to observed amenities. However, this assumption is not necessary, and a more generaltreatment in which a vector of unobserved amenities also exists is developed by the companion paper (Bellet al., 2023).

11

I.B

Subsequent Research with this Method

While the empirical application contained in this paper shows one way that the proxymethod can be useful, several other works citing earlier drafts of this paper have now also
used this method to yield important economic insights.
Most applications of the method to date have been in labor economics. Folke andRickne (2022) apply the method to a nationally representative Swedish dataset to find thatlabor markets do not seem to compensate employees for the risk of sexual harassment.18
Burbano et al. (2023) investigate how the premium that meaningful work commands might
contribute to gender inequality. Similar to Figure 6 of this paper, they present estimatesfrom seven different methodologies of the pay premium for meaningful work, and the othersix of the seven indicate that meaningful work is a disamenity.19 Using the price estimate formeaningful work that they dub the “Bell Compensating Pay,” they find that the premiumcommanded by jobs with more meaning contributes 9.5% of Sweden’s gender pay gap.
Applications of the method have also proven fruitful outside of labor economics. Bell etal. (2023) apply the method to the study of residential amenities with a migration-based
measure adapted from Sorkin (2018) as the proxy for latent neighborhood quality. Finally,
complimentary work by Willen and Zhang (2023) adapts an analagous single-index framework to the study of mortgage discrimination, but they develop an intriguing alternative
procedure for estimation that results in set, rather than point, identification.

II

Economics of Compensating Differentials

My conceptual framework follows at a high level the model of Rosen (1986) in the Handbook of Labor Economics.20 To emphasize the problem of ability bias, I introduce thestandard continuous Rosen model first in terms of workers that are homogenous in theirability (but differ in preferences). I then introduce ability as the second level of exogenous
heterogeneity.21
18 Interestingly, this finding fits with their other analyses, which suggest that women in general sort awayfrom jobs that confer a high risk of sexual harassment, which leads moreso to gender seggregation in thelabor market than a pattern of compensating differentials for the disamenity.
19 The authors also illustrate that the method assigns the expected sign to other amenities in their data,
including flexible work times (positive), physical exhaustion (negative), and influence over work structure(positive).
20 A more mathematical treatment of the closed-form integral solution to a binary version of this model
can be found in Online Appendix Section B.
21 As a more realistic (but substantially more complex) alternative to this two-step model, Han andYamaguchi (2015) derive conditions under which a stable worker-firm match arise when markets form underboth dimensions of heterogeneity simultaneously. A simulation of a similar simultaneous matching process
is also included in Appendix D of Bell et al. (2023).

12

II.A

Relative Taste Heterogeneity

To start, workers are equally productive and differ only in how their compensation is split
into various forms. To illustrate the model, I will assume compensation is split between
wage w and only a single amenity z, but the framework generalizes to substitution across
many amenities.22 Index worker i’s valuation of the amenity by θi and firm j’s cost ofproviding it by Ωj . Worker utility is given by u(wi , zi |θi ) and firm profits by π(wj , zj |Ωj ).
The equilibrium price will be determined by the distributions of θ and Ω.
Market clearing is defined as a matching of workers to firms and a wage and amenity
paid out in each job. Efficiency dictates workers with the highest valuation of the amenity
are matched to firms with the lowest cost of producing it: perfect assortative matching on
θi and Ωj . Each worker-firm pair determines a level of income and amenity that is Pareto
optimal, which means that the (w, z) coordinate chosen will be at a tangency of the worker
and firm indifference sets, specified by u and π. Famously, all workers’ and firms’ indifference
sets of u and π are “kissing” in equilibrium. For the matching of workers and firms to bestable, the (w, z) coordinate chosen by a particular pair will also be one that is less desirable
to neighboring workers and firms with similar valuations, relative to what those neighbors
have chosen.23 With a sufficiently continuous distribution of θ and Ω, the relative location
choice of each pair is unique and constitutes the unique equilibrium within the market.24
From the perspective of a single participant who takes other participants’ choices asgiven, the convex combinations of w and z that have been agreed upon function as a budget
set. The local slope of this set – how much wage must be given up to obtain a small amountmore amenity – is the local price of the amenity in wage units. A single worker’s problem
can be rewritten as maximizing u(wi , zi |θi ) subject to the non-linear budget constraint thatwi and zi are points along a particular frontier. All workers at this point must lie on thesame frontier, because productivity does not yet differ.

Firms, too, maximize profits

subject to the feasible set of options given by this frontier. This is a partial-equilibrium
interpretation of the frontier, but if many workers or firms changed their choices, the shapeof the frontier could change in the new equilibrium. The first two panels of Figure 3 illustrate
the relationship between the slope of the frontier and the distributions of preferences andfirm costs that have contributed to its shape.
22 Although wage refers to an hourly concept, the logic of the model would also generalize to different
earnings measures, such as yearly earnings among a group of full-time salaried workers. The model isintended to explain the worker’s labor supply decision between different types of jobs, rather than of how
many hours to supply.
23 Figure A.6 illustrates that if the frontier deviates from this condition, it is not a stable equilibrium.
24 While the location choice of all pairs relative to each other is unique, there is a level parameter that is
not pinned down by this model. It can be thought of as homogenous bargaining weights that determine theway productive outputs are split between workers and firms. I take that process as exogenous.

13

Figure 3: The “Kissing Equilibrium”
Panel A: Worker and Firm
Tangency

Panel B: The Frontier

Wage W
Wage WΨ local slope:
uw/uz or πw/πw
w1+Ci
w1+Bj
w1

ui=w1+Ci(z)πj=w1+Bj(z)Amenity Z
Dirty
(z=0)

Clean
(z=1)

Amenity Z
Panel C: Multiple Frontiers

Notes: This figure graphically represents the equilibrium between labor supply and demand discussed in
Section II, and illustrates how the case of a binary amenity can be thought of as a discretization of acontinuous model. In Panel A, an indifference curve for a particular worker is drawn in green, such that Ci
represents the amount of wage required to make the worker indifferent with a one-unit change of amenity(a change from 1 to 0, following the binary treatment of Online Appendix B). The blue curve depicts theanalogous level-set of profits for the firm that has matched with the worker. The particular level sets ofutility and production are chosen such that they meet at a tangency. Panel B shows the tangency for thispair and for an additional worker-firm pair. The frontier ψ is drawn in black such that it passes through thetangencies for all pairs, and such that ψ is tangent to each worker/firm indifference curve (which restricts
which level-sets each pair chooses). Appendix Figure A.6 illustrates that if a worker-firm pair chose a levelof wages and amenities such that the frontier did not satisfy the latter tangency condition, then the match
would not be stable. Finally, Panel C demonstrates the same sorting equilibrium for multiple sub-markets
that are ordered by how much compensation they entail. For a single worker of a particular ability level η,
the corresponding ψ frontier constitutes the set of obtainable jobs, similar to a budget set. The local slopeof ψ is viewed by the worker as the price of the amenity, in wage units.

14

II.B

Absolute Heterogeneity

Finally, I extend the model to incorporate the fact that some jobs afford better compensation than others. To incorporate this feature, I model the amount of total compensation theworker can obtain as heterogeneous and determined exogenously, perhaps by factors determined prior to labor market entrance such as intelligence, motivation, and the unobserved
facets of job search. Firms differ in their exogenous staffing needs and pay out more totalcompensation, composed of wages and amenities, in accordance with what type of applicant
they seek to attract.25 The multiple Rosen frontiers are shown in Panel C of Figure 3.
The critical modification is that total compensation, which I have denoted by η, is nolonger the same for all workers.26 Viewing η as pre-determined, the equilibrium can still
be found by sub-dividing the entire labor market into sub-markets indexed by η. Markets
indexed by the highest η tend to contain the workers with the most productive abilities,
matched with firms that most value such human capital. Within each η-level sub-market,
wage and amenity are determined in accordance with the Rosen framework in which all jobs
entail an η amount of total compensation, but in different forms. The amenity price is now
local to the sub-market. For a sub-market of compensation level η, the price is governed bya frontier that I will denote as a function:1 w and z entail total compensation valued at ηψ(w, z|η) =0 otherwise
The ψ function is strictly monotone in both arguments, such that the frontiers of different
compensation levels do not intersect each other.

II.C

Stochasticity of the Offer Curve

Decompose the determinants of worker i’s compensation ηi into an observable part hi andan unobservable part εi . The unobserved determinants of compensation in ε can include
unmeasured skills, but also random search and matching frictions which result in observably
identical workers obtaining better or worse jobs.
To the best of my knowledge, all prior work on estimating market prices of amenities has
made the assumption that wages are stochastic (caused by a variable like ε) but amenities
are not.27 To the extent that search frictions exist in the labor market (Goldberger, 1984;
25 A more complex model of the labor market would allow firms to have some elasticity of substitution
between skilled and unskilled labor. However, the goal of the present paper is not to understand the factors
determining why higher-skilled workers earn more than lower-skilled workers, but rather the income-amenity
choices faced by workers of a similar skill level. Thus, I take the skill premium as determined outside of themodel.
26 Differences in η can capture not only productivity, but also to some degree due to luck or search frictionsin the Mortensen (2003) sense.
27 When wage is regressed on amenities and proxies of the offer set, the regression typically has an R2 farless than 1 (Hwang et al. 1992). This can only be explained by measurement error in wage or a structural

15

Mortensen, 2003), it does not seem plausible that they would affect wages but not amenities.28 A key deviation of my model from the literature is that I treat total compensation –and thus both wage and amenity – as stochastic.
This assumption relates to what can be learned from conditioning on workers’ wages
and amenities. In the model consistent with the OLS approach, there is agnosticism about
whether two workers with the same wage and amenities actually share the same latent offer
of total compensation for which we seek to control. (This is because at a given level ofamenities and offer sets, latent variables are still assumed to influence wages.) In my model,
if two workers obtain the same wage and the same amenities, I view them as necessarily
sharing the same latent offer set. Therefore, a new assumption that I introduce into theempirical literature on job choice is that each value of wage and amenity should be consistent
with only one level of total compensation. In the proof of Section III, this translates to anassumption that η can bet written as a function of wage and amenities.

II.D

Equilibrium Exclusion Restriction

Among workers of a total compensation level η, h must be independent of worker and firm
tastes, θ and Ω. Formally:(h ⊥ θ)|η and (h ⊥ Ω)|η(1)

There is some redundancy here in that the condition on the relationship with θ is reallythe same as the one on the relationship with Ω. Under the sorting framework, at job k,θk = Ωk so (h ⊥ θ)|η is interchangeable with (h ⊥ Ω)|η. However, I state both conditions toemphasize that one can evaluate the exclusion restriction from either the firm’s perspectiveor the worker’s perspective – and an argument in violation of either is sufficient to rule out
a candidate h variable.29
Finally, the parameters θ or Ω in Condition 1 can be replaced with the joint distributionof w and z to generate a more reduced-form statement of the exclusion restriction.
((w, z) ⊥ h)|η(2)

Identification relies on a variable that is informative about how high of a compensation
frontier the worker lies on (η), but adds no f urther information about the location of
the worker along that frontier. Put another way, h would be irrelevant to w and z if theerror term ε. For the parameters of this equation to be identified by OLS, it must be the case that thestructural error causes wage, but is unrelated to amenities.
28 In Appendix D.A, I delineate the assumptions that support my view that total compensation is stochastic
versus the typical assumption that wage is stochastic and amenity is not.
29 From the worker side, within the η-level workers, those that have high h can’t have different valuations of
the amenity (θ) from those that have low h. From the firm side, among firms that pay η-level compensation,
those that employ high-h workers can’t have different valuations of the amenity (Ω) from those that employ
low-h workers.

16

researcher could observe and condition on η.

II.E

No “bad controls”

To see why my assumptions are weaker than those of the typical OLS approach, note thatthe typical approach requires the researcher to select control variables for ability. I will showthat any ability control that satisfies OLS exogeneity when regressing wage on amenities
also satisfies Condition 2.
The typical amenity pricing approach seeks to estimate the joint distribution of w and z
conditional on η, which I denote as (w, z)|η. For instance, this may be linearly approximatedby a regression of w on z with controls. A control is a bad control if it can cause bias,
e.g. if adding it to an already-identified model would cause identification of the structuralparameters to be lost.30 Defining (w, z)|η as the identified distribution of interest, then, agood control would have the property that adding it to the conditioning does not change
the w-z relationship of interest:(w, z)|η, h = (w, z)|η
Rewrite the “good control” condition in terms of PDF’s:
fwz|η,h = fwz|η
Re-writing the left-hand side:
fwzh|η
= fwz|η
fh|η
Re-arranging,fwzh|η = fwz|η fh|ηBy definition of independence, I arrive back at Condition 2:((w, z) ⊥ h)|η(3)

In the typical analysis of wage on amenity with ability controls, Condition 2 is necessary
to include h as a control variable. The only difference is that with the standard regression
approach, I would also require the rest of the full set of ability controls. Therefore, theproposed estimator entails a weakening of the assumptions required by the OLS approach.
30 Angrist and Pischke (2009) write: “Some variables are bad controls and should not be included in aregression model even when their inclusion might be expected to change the short regression coefficients.
Bad controls are variables that are themselves outcome variables in the notional experiment at hand. Thatis, bad controls might just as well be dependent variables too. Good controls are variables that we can thinkof as having been fixed at the time the regressor of interest was determined.”17Any single variable allowed to be included as a control for OLS should also be a valid hvariable for my approach.
II.E.1

Pre-Determination

Related to the idea of “bad controls,” one can conceptually test whether an h variable isvalid by asking whether it would change in response to the equilibrium amenity price. Validh variables should not respond to prices.
For example, following Goldin and Katz (2011) suppose that pharmacists, relative tolawyers, tend to be paid in family-friendly amenities. Workers sort into occupations on
valuations of that amenity. Although score on a law-school entrance exam might be relevant
to workers’ levels of compensation, it would not be a valid h variable. It is not pre-determinedbecause if the labor market changed such that lawyers were also paid in family-friendly
amenities, one would expect a very different group of workers to move in to law. If anything,legal knowledge would be determined in response to the wage-amenity equilibrium. In
contrast, a skill that is more widely applicable to a range of occupations (such as a cognitive
aptitude measure) would not be expected to respond to the outcome of the wage-amenity
sorting equilibrium.

II.F

AFQT as an h variable

I argue that score on the Armed Forces Qualification Test (AFQT) is correlated with thelevel of the wage-amenity frontier, but is independent of workers’ locations along their
particular frontiers. The AFQT score is the primary summary score from the Armed Services
Vocational Aptitude Battery, which was administered to respondents of the NLSY79 in 1980.
As the purpose of the test is to measure basic skills that are applicable to a wide rangeof vocations, the score by construction should be independent of how employers locatedat various positions on a wage-amenity frontier view a worker. Although AFQT is in nosense “innate,” any effort by a worker to increase his or her AFQT score should result only
in a movement to a higher compensation frontier, rather than a movement along a given
frontier. Perhaps the most pressing concern is that the decision to invest in AFQT-relevant
skills could for some reason be correlated with a preference for wage or for certain typesof amenities. For instance, such an argument might be that workers who have a stronger
affinity for certain amenities invest more into training for general-purpose skills. Still, suchan argument is hard to rationalize with utility-maximizing individuals.
In a more reduced-form sense, AFQT is plausibly a noisy proxy for productivity, in which
the noise is uncorrelated with preferences. Heterogeneity in income versus amenities among
η-level workers and firms would be determined not by AFQT but rather by workers’ andfirms’ heterogeneous valuations of the amenity. The key is that once we’ve conditioned on
η, AFQT should be irrelevant to both the firms’ and workers’ problems.
18

Finally, several foundational papers in labor economics have used AFQT score fromthe NLSY as a noisy measure of worker productivity. Employer learning and statistical
discrimination have been explored by Farber and Gibbons (1996) and Altonji and Pierret(2001), who used AFQT as a measure of worker skills that are unobserved to the employer.
In a different vein, Neal and Johnson (1996) used AFQT as a “measure of basic skills
that helps predict actual job performance,” noting in particular its usefulness of measuring
skills present prior to labor market entrance. The importance of AFQT in explaining thewage gap has often been cited as evidence of the racial “skill gap.” A central assumption
in prior literature has been that AFQT measures general-purpose skills, not preferences.
For instance, although many interpretations of the findings of Neal and Johnson (1996)
have emerged that do not involve a racial “skill gap,” no arguments have been made that
controlling for AFQT is revealing a “preference gap” for job amenities, which is precisely
what would be needed to invalidated to estimate the neoclassical model of compensating
differentials.

III

Non-Parametric Identification

In this section, I prove that the Rosen frontier can be non-parametrically identified underthe equilibrium exclusion restriction of Condition 2, in which ability proxy h is assumed
irrelevant to wage-amenity sorting conditional on true ability η. In the next section, I apply
the non-parametric result to a data-generating process assumed to be linear, in which I
discuss issues of potential bias.

III.A

Object of Interest

Denote wage by w, amenity by z, and total compensation by η. The goal is to learn what
wage and amenity values are associated with each level of total compensation, denoted in
Section II as the function ψ(w, z|η).

III.B

Assumptions

I make the usual standard assumptions: an IID sample of w, z, and h; and, all variables
are assumed continuous. The substantive assumptions are:
1. Conditional independence of h: ((w, z) ⊥ h)|η
2. Monotonic relevance of h: E[h|η] is everywhere strictly monotone in η
3. η is degenerate given values of w and z; i.e., there exists a non-stochastic function gsuch that η ≡ g(w, z)

19

Assumption 1 is what I have referred to as the “identifying assumption,” although it was also
an identifying assumption for the typical approach that controlled for measures of the offer
set. Assumption 3 is the assumption that enables identification by the proxy variable. The
global strict monotonicity of Assumption 2 can be relaxed; Online Appendix F discusses
identification under a more local form of monotonicity.

III.C

Proposition

Under these assumptions, ψ(w, z|η) is identified for all levels of η.31

III.D

Estimator

Define the random variable ĥ as an approximation to E[h|w, z]. For simplicity, ĥ may simply
be a conditional mean of h in given bins of w and z.32 Denote with subscripts ηi as quantile
i of the unobserved variable η and ĥj as quantile j of the constructed variable ĥ. For allobserved w and z, ψ(w, z|ĥj ) approximates ψ(w, z|ηi ) for some unknown quantiles i and j.

III.E

Proof of Identification

The first step is to note that the constructed variable ĥ = Ê[h|w, z] can be equated, inprobability limit, with E[h|w, z, η]. This is because η is assumed degenerate given w and z,
so adding it to the conditioning is redundant (Assumption 3).
In the second step, I rely on conditional independence of Assumption 1 to equate
E[h|w, z, η] from above with E[h|η]. This is intuitive, since the wage-amenity split is byassumption not related to h.33
At this point, I have shown that under the conditional independence assumption, ĥ approximates E[h|η]. My last assumption is that E[h|η] is strictly monotone in η (Assumption2). If this is the case, then each quantile of the control function corresponds to exactly one
quantile of η. Conditioning on an arbitrarily small range of the control function makes thecorrespondence to the level of η arbitrarily close.
Quantiles of ĥ can be mapped to quantiles of η as follows. Consider a worker i withobserved wi and zi and an unknown level of compensation, ηi . The researcher is interested
in learning what other observed values of of w and z are associated with this particularamount of total compensation and are thus also in the worker’s choice set; I denote thisby the function ψ(w, z|ηi ). To find the quantile j of ĥ such that ψ(w, z|ĥj ) approximates
31 To clarify, the units of η are to be viewed as a nuisance parameter. The goal is to learn the values of w
and z associated with the same η, but will remain agnostic about the cardinal units of that η.
32 Alternatively, optimal kernel estimators might be considered for efficiency, which is not the present
focus.

33 A full derivation of this step is provided in Online Appendix C.

20

ψ(w, z|ηi ), all that one needs to do is find the ĥj such that ψ(wi , zi |ĥj ) = 1. Since ĥ will
be defined on any point for which w and z are observed, for all observed w and z, ψ(w, z|ĥ)
identifies ψ(w, z|η) for all levels of η.

IV

Estimation in the Linear Model

For the purpose of discussing potential biases, it is now useful to restrict the model to one
in which price can be summarized by a single parameter. The non-parametric estimatorentails nothing more than plotting conditional means of the h variable by bins of wage
and amenities. The linear proposed estimator is best understood as a special case of that
non-parametric estimator, in which conditional expectations are allowed to be replaced with
their linear counterparts and can thus be estimated by linear regression. Relative to thenon-parametric framework, I will introduce two additional assumptions of linearity: linearityof the Rosen frontier and a linear relevance of h to η.
In the non-parametric framework, the goal was to identify a function of w and z given
unobserved η. Here, I will take the object of interest to be the coefficient on z from ahypothetical regression of w on z controlling for η. That is, I will assume E[w|z, η] is linear
in z, with homogeneous parameter β, such that each Rosen frontier can be written of thelinear form:w − βz = η(4)I have omitted a scale coefficient on η as it makes no difference since η is unobserved andI have normalized the price of wage to 1. The content of Equation 4 is simply that I needto assume that conditional on η, the values of w and z form a line if I am to summarize thisrelationship with a single parameter. The second assumption of linearity is more nuanced.
In the non-parametric framework, I assumed E[h|η] was strictly monotone in η. Now, I
will strengthen monotonicity to linearity. In other words, E[h|η] can be written γη for some
unknown γ.
With these additional assumptions, one can modify the proof of identification in Section
III to replace all conditional expectations with linear regressions. The table below provides a side-by-side comparison of the non-parametric and linear estimators. I have alreadyshown graphical intuition for the linear approach using predicted values in Figure 2; thenon-parametric approach simply uses predicted values that are not required to be linear
functions.
Non-ParametricLinear1. ĥ = Ê[h|w, z]

1. ĥ = Eˆ∗ [h|w, z]

2. Plot w on z at various quantiles of ĥ

2. Regress w on z controlling for ĥ

21

Figure 4: Illustration of Bias

Wage

Wage

Safety(a) Unbiased

Safety(b) Biased

This figure discretizes a hypothetical labor market into three segments of high, medium, andlow compensation, represented by linear Rosen frontiers drawn as heavy black lines. Colors
are used to depict levels of observed h; in ascending order blue then red then green (circles,
squares, diamonds). No bias is present in Panel A because within each η-level frontier, his distributed independently of w and z. When the ability channel is normalized to this hvariable, the estimated price would lie on-top of the true frontiers as desired. In Panel B,
additional observations have been added at the edges of the frontiers such that h is clearly
not independent of w and z conditional on the frontier. Matching the example discussed in
Section IV.A, on each frontier workers with the highest h now also have the highest z, andlikewise those with the lowest h have the highest w. The positive conditional covariance of hand z leads to a rotation of the price estimate, shown in dashed lines. The slope estimate ismore negative than the true one. In other words, the estimator would suggest that workers
give up more wage for this amenity than is truly the case.

IV.A

Bias when (w, z) 6⊥ h|ηIn the linearized model, the bias in β̂ takes the sign of cov ∗ (w, h|η). Online Appendix I
contains a formal proof, but Figure 4 conveys the key intuition in the linear model. No bias
occurs when the h variable is evenly distributed along each Rosen frontier. To the extent
that high-h workers bunch on one end of the frontier, they will pull the estimate with them.
For instance, perhaps one suspects that a candidate h variable is positively correlatedwith workers’ preferences for amenities (relative to wage). The excess bunching of high-h
workers at high amenity levels would cause us to estimate that this is a more costly amenity
than it actually is. This argument matches what is depicted in the figure, as would a story inwhich h was a skill that was particularly valued in high-amenity jobs (relative to high-wagejobs).

22

IV.B

Finite-Sample Inference

In general, the linear proposed estimator is vulnerable to a finite-sample problem similar tothat of weak instrumental variables whenever the relationship between h and w is difficult
to detect empirically. The problem is the result of reporting the price of the amenity perunit of wage, which puts the wage coefficient into a denominator.
Recall the ratio of coefficients estimator for the price is − π̂δ̂ from a regression of h on w
and z:Eˆ∗ [h|w, z] = δ̂w + π̂z
Whenever δ̂ may dip close to 0, the ratio of coefficients explodes, causing price estimatesto skew large. In light of the potential for bias in finite samples, I report Anderson-Rubinconfidence intervals when appropriate, following the suggestion and approach of Andrews
et al. (2019).

IV.C

Measurement Error in w and z

This estimator is affected by measurement error in a very different way from the typical OLS
approach. Like the finite sample bias, the bias from measurement error is best understoodfrom the ratio-of-coefficients approach, which runs the regression Eˆ∗ [h|w, z] = δ̂w + π̂z andtakes the ratio − π̂δ̂ .
Classical measurement error in h causes no particular bias on π̂ or δ̂. This is a key
advantage of my estimator; noise in the dependent variable of a regression only causes
imprecision but not bias. (Although, as previously discussed, this can itself cause finitesample bias.)
Measurement error in a regressor, on the other hand, causes the coefficient on that
regressor to be attenuated. If w is measured with noise, then δ̂ is too small, so the estimatedprice of the amenity is too large in magnitude. If z is measured with noise, then π̂ is too small,and the price of z is too small. Because all components of compensations are regressors,
measurement error in any piece of compensation will simply make it appear less importantor valuable. Of course, because measurement error in wage and amenity have the potential
to balance each other out, there is also the case in which both variables are measured witha balanced amount of noise such that the resulting price estimate is unbiased. A fuller
treatment of measurement error is given in the reduced-form statistical model of Online
Appendix G.

23

V
V.A

Price Estimates

Data

The primary data sources for the analysis are the NLSY79 and O*NET. The analysis sample
is NLSY respondents employed in 2012 linked at the occupation level with O*NET measuresof amenities.
The NLSY79 is a longitudinal survey dataset constructed by the Bureau of Labor Statistics. The first wave, administered in 1979, sampled 12,686 men and women between theages of 14 and 22. Subsequent survey waves were administered annually until 1994, andonce every two years since. The NLSY contains three sub-samples. The primary sample(6,111 people) is “designed to represent the non-institutionalized civilian segment of young
people living in the United States in 1979 and born January 1, 1957, through December
31, 1964.” The NLSY contains two additional samples; the first (5,295) is a sub-sample ofracial minorities and economically disadvantaged people, and the second (1,280) is designed
to represent those serving in the military as of September 30, 1978. To maximize power, I
include all three samples.34
I focus on data in 2012, as the occupation data has changed across years. By 2012,
only 7,300 of the initial NLSY respondents were retained. Of the 5,386 not interviewed, thebiggest driver has been administrative decisions to reduce the sizes of the non-representative
sub-samples.35 My final analysis sample consists of 4,868 NLSY respondents who reported
being employed in 2012, when they were aged 48-55. This number excludes 152 individuals
who I drop due to missing job amenity fields. To be employed, the respondent must haveworked at least ten hours per week and at least 9 weeks since the date of the last interview,
or be self-employed. Although I conduct the main analysis on this broad sample, I also test
robustness to selecting on the full-time employed population. In the analysis of demographic
pay gaps, I divide the NLSY by gender and by race. Gender and race both correspond to
the respondent’s data in 1979, at the start of the survey, as determined by an interviewer.
Sex and gender were not coded separately at this time, so I use them interchangeably. The
racial/ethnic categories are as follows: “hispanic,” “black,” or “non-black/non-hispanic.” For
the analysis by race, I will compare the black group to the non-black/non-hispanic group,
which I will at times refer to by the approximate term “white” for brevity.
For additional results on income inequality by parent background, a PSID sample wasconstructed to mirror the sample definitions of the NLSY. The PSID sample contains 929
workers in 2013 linked to their parents. For each child, I define a measure of parental income
34 In the final analysis sample, only 6 workers list Armed Forces (Census code 984) as their occupation in2012.
35 84% of the military sub-sample was dropped after the 1984 survey, and in 1990 31% of the disadvantaged
sub-sample was dropped (poor whites). After the 2,722 individuals dropped for one of these reasons, themost common reasons for attrition were refusal (903), can’t locate (466), deceased (689), other (481), anddifficult cases (125).

24

to be the father’s income averaged over all years for which the father was between 30 and
50. Because of the requirement that I observe income for both workers and their fathers,
this sample of workers is substantially younger than the NLSY one, between ages 30 and
40.
O*NET, which stands for Occupational Information Network, is a database of occupational characteristics maintained by the US Department of Labor. The data are derived
primarily by occupational surveys and supplemented in some cases by occupational experts.
I draw primarily on the O*NET Work Context table, which includes attributes that workers
might plausibly have varying preferences over. Examples include how regular the scheduleis, how often minor cuts and burns arise on the job, whether the work is indoors or outdoors,or how much freedom there is to make decisions. As the scales of most of these measures
are difficult to compare, I translate income and all amenity measures to standard deviation
units across workers in the sample. Table A.1 describes the process by which several amenity
measures were constructed from O*NET.
I make two key deviations from the logic of the simple conceptual framework in which
workers choose jobs indexed by a wage and amenity level. First, whereas the canonicalframework allows workers to choose particular jobs, in my data many amenities are only
observed at the occupation level via the O*NET occupation survey. The difficulty of measuring amenities at the job rather than occupational level has been discussed in prior work
(e.g., Goldin (2014); Denning et al. (2019)). Due to this limitation, I aggregate the entire
analysis to the occupation level, rather than job level, and weight by the number of workers in each occupation.36 I also show robustness of the final results to estimation without
aggregation. Collapsing the data to occupation level entails a loss of power. After mapping
the O*NET SOC occupation codes to the same Census format as is used in NLSY in 2012,
I am left with only 397 occupations that are represented in the NLSY in 2012 (451 acrossall years). This means that 397 is the effective sample size, or number of clusters, for mostof the analysis.
Second, while the basic framework considers the trade-offs workers face between wage
and amenities, in practice most workers in the sample do not report being paid an hourly
wage (Figure A.2). Although it is possible to impute an hourly wage based on workers’
estimates of hours worked, to more accurately mirror the level at which most workers aremaking decisions, I use yearly labor income rather than hourly wages. Labor income in
the NLSY is defined as total pre-tax income from wages and salary in the past calendar
year, but if a worker holds more than one job then it is aggregated over all jobs. The
amenities assigned, however, will correspond to the worker’s “primary” job. Assuming thatany additional jobs held by the worker have similar amenities to the primary job, then
36 Appendix Table A.2 contrasts the findings of the typical OLS approach on data at the individual versus

occupation level; even with OLS, aggregation of the entire analysis to the occupation-level does seem toremove some, but not all, of the wrong-signed bias.

25this data definition can be viewed as the relevant yearly income and typical amenity levels
consumed by the worker.

V.B

Functional Form & Relation to Prior Approaches

As a first look at empirics, Figure 5 compares the OLS approach with the proposed approach,overlaying linear estimates on-top of their non-parametric counterparts. Similar to Figure 1,
the OLS approach in Panel A entails regressing income on safety, controlling for the ability
proxy of AFQT by way of terciles. Within terciles of AFQT, the income-safety relationshipis upward sloping. In contrast, the proposed approach in Panel B uses precisely the samedata but plots mean AFQT by bins of income and safety. Despite identical underlying data,
the finding is the opposite: within each tercile of mean AFQT, which should be thought ofas the empirical stand-in for the offer curve, the income-safety relationship is now downward
sloping, implying that a trade-off exists in the labor market between income and safety.
Just how steep is the trade off, and is it well characterized by a single parameter? My
answers to these questions are limited by the precision of the data. In Panel B, for instance,
although a downward trend is apparent, some bins fall off the trend. This is likely due tosampling variation. In Panel C, a formal statistical test is applied to do inference on thegraph. Because every bin represents a sample mean of AFQT, I can apply a standard twosample t-test for difference in means to help determine which regions of the graph correspondto different offer curves. I have arbitrarily picked the center point as the reference point, bywhich all other means are tested. The test constructs a graphical confidence interval for the
price, allowing the researcher to evaluate potential non-linearities around the center point.
Although the graphical test can reject many upward-sloping patterns that the wage-amenity
frontier passing through the center point might have taken on, there is still a great deal of
uncertainty about the exact shape of the frontier. Due to the low level of precision that canbe obtained from this relatively small sample, for most of the remainder of the paper, I simply
estimate linear average prices rather than attempting to gain non-parametric precision.37
In a linear model, Figure 6 compares several of the typical strategies with the proposed
estimator. In this case, I define the Mincerian estimator to be the regression of wage on theamenity with controls for education, AFQT, age, and year. I study the price of amenities
one-by-one in relation to income. For instance, examining the amenity of safety, the raw,
Mincerian, and individual fixed effects estimators all produce price estimates indicating that
37 Related to the problem of imprecision, the potential for finite-sample bias when summarizing the datalinearly can also be detected in Figure 5. Because my data has sampling variation, some of the points inPanel B appear to lie off trend. When the viewing this graph, the noise in the measure of the offer curve
level is apparent. Now, consider a regression of income on safety controlling for the estimated level of theoffer curve, which contains sampling variation. For any level of the noisily estimated offer curve, there is aweaker relationship between wage and amenity due to the sampling variation, so the linear estimate of theslope of interest looks more like a vertical line that it should. This is the graphical interpretation of thefinite-sample bias that was shown algebraically in Section IV.B.

26

Figure 5: Non-Parametric & Linear(b) Proposed Estimatorincome(a) OLSsafety(c) Proposed Estimator, 95% CI

Notes: Panel A is a binned scatterplot of income by bins of safety, separately by terciles of
AFQT, in the NLSY analysis sample at the occupation level. Green diamonds are workers
with the highest AFQT, then red squares, and blue circles are the lowest-AFQT workers.
The black line is the linear best-fit: on average, workers with more safety also earn moreincome, even within skill bins. Panel B plots mean AFQT by income and safety; thesemeans have been discretized into terciles for visualization. The black line is the linear bestfit of income on safety controlling for the bin’s mean AFQT. Panel C is similar to PanelB, except the dots are shaded in the following way. First, the center dot is made dark to
indicate it is the reference point. Then, two-sided t-tests are performed to test whether themean in the reference bin is significantly different from each of the other bins. Bins in whichmean AFQT is indistinguishable at the α = .05 level are colored in blue, and the rest in
grey. For all panels of this figure, income and safety are in percentiles rather than standard
deviations. All panels of this figure use the odd number of 9 bins so that a center dot canbe chosen in Panel C. Additional sample details are found in Section V.A.

27

Figure 6: Raw, Mincerian, Fixed-Effects, and Proposed EstimatesRaw
safetyMincerian
Individual FE

decision
freedom
Proposed
non-profit

regularschedule
-.6

-.4

-.2

0

.2

.4

Compensating Differential Paid in Income

Notes: Each data point on the bar graph is a separate (bivariate) regression of the price of
the amenity versus wage, controlling for no other amenities, in the NLSY sample at the occupation level. Income and all amenities are in standard deviation units. Mincerian includes
indicators for years of education, exact AFQT score, and age. 95% CI’s are calculated at
the occupation level, and are identification-robust for the proposed estimates using AFQT.
All series use the NLSY sample in 2012 except for the individual fixed effects specification,
which extends the years to 2004-2014. Additional sample details are found in Section V.A.
the average price of safety is negative (it is a disamenity), similar to what was reported byBrown (1980). On the other hand, the proposed estimator for the price of safety when
the h variable is AFQT lies on the other side of 0 and yields a result contrary to all otherestablished methods in observational data. The point estimate indicates that on average,
workers must give up .31 standard deviations of income to obtain a typical job with astandard deviation more of safety.38 Similar descriptive compensating differentials can befound for jobs that have more regular schedules (.29 SD less income), those with more
freedom to make decisions (.07 SD less income), and jobs that tend to be in the non-profit
sector (.42 SD less income). Next, I turn to examine how results from the proposed estimator
compare when proxies of workers’ productive abilities other than AFQT are used.
38 As discussed in the introduction, this is not precisely the “price” of safety, as it does not control forother amenities. The price of safety would describe the trade-off holding all other attributes fixed. Rather,
this estimate describes the trade-off between income and a typical safer job.

28

V.C

Sensitivity to h variable normalization

A strength of the NLSY is its preponderance of measures of skills. There is no way to prove
empirically that any potential h variable fits the identification assumption. Whereas eachpotential ability proxy may always be subject to its own nuanced critiques as to why it may
not fit the identification assumption, an opportunity exists to compare results from manydifferent independent measures of skill. If very different proxies for the ability channel yield
similar results, this can lend some credibility.
V.C.1

Data on h variables

In section II.F, I argued that a cognitive test such as the AFQT, though by no means“innate,” may be an ability proxy not contaminated by wage-amenity sorting. This would
be plausible so long as workers do not systematically manipulate their AFQT in order togain particularly more of either wage or amenity, rather than total compensation. The
AFQT was administered to NLSY respondents in the 1981 survey year.
A variable that may be more pre-determined is the worker’s height, which is surveyed
in the NLSY in several years including 2012. The positive relationship between heightand earnings, which holds even in developed nations, has been well documented though
its source is a topic for debate. The cause of this association does not seem to be thatheight is a skill that is useful in certain physically demanding jobs – i.e., it does not seemto be the case that warehouse firms compete for taller workers who can reach higher boxes.
Evidence against this explanation is presented by Persico et al. (2004), who show that theheight premium is entirely distinct from a worker’s contemporaneous height once height
during adolescence is controlled. Thus, the returns to height must be running through anunobserved variable correlated with adolescent height. Persico et al. (2004) propose thatthe skills and confidence stemming from increased social and athletic experiences duringhigh school may partly be at play. However, more recent work by Case and Paxson (2008)
makes a convincing case that taller adults earn more because they have higher cognitive
skills. Since physical development is known to be influenced partly by in utero and earlychildhood environmental factors such as disease and nutrition, such environmental factors
would also act in the same direction on cognitive development. Whichever combination ofsocial, cognitive, or even taste-based discriminatory explanations one wishes to believe, theevidence indicates that height proxies for an ability channel that is likely to be irrelevant
to where workers locate along the wage-amenity frontier. To better isolate the componentof height accounted for by environmental rather than genetic factors, I follow the recent
literature by residualizing height on sex.39
Other plausibly exogenous variables include education and measures of what might becalled social skills. Education, measured in the NLSY as highest grade completed by 2012,
39 Results are similar when height is also residualized on race.

29

Table 1: Correlations Among Income & Proposed h Variables

IncomeYears Educ.
AFQT
Self-Esteem
Mastery
Locus of Control
Height (res. sex)

Income10.36850.35790.18710.1816-0.1501
0.066

Educ.

AFQT

Self-Est

MasteryLoc. C.

Height1
0.54260.28030.2392-0.21170.11851
0.31750.2084-0.3095
0.14861
0.2922-0.28140.05931-0.1605
0.05771-0.0492

1

N otes : This correlation matrix summarizes the main NLSY analysis sample with all variables non-missing (4,257 workers). Additional sample details and definitions of h variables
are contained in Section V.A.
is a variable often used as one of several controls for ability in the labor economics literature,and has often been included as a control when researching compensating differentials. As
described in Section II.E, any valid ability control from the typical OLS or “Mincerian”
specification is a valid h variable here. I also use three scores contained in the NLSY fromtests that have been referred to by previous literature as “social” or “non-cognitive” skills
that are predictive of compensation (e.g., Heckman et al. (2006); Deming (2017)). Thesevariables are self-esteem, mastery, and locus of control. The self-esteem measure, pioneeredby Rosenberg (1965), is based on a 10-item survey that gauges agreement with statements
about attitudes toward oneself (e.g., “I have a number of good qualities.”). Mastery is asimilarly constructed variable based on a survey developed by Pearlin et al. (1981) to study
factors that mediate stressful life events such as job losses; it is a self-reported measure ofa worker’s ability to solve problems that affect him or her. Finally, locus of control is avariable constructed from an abbreviated version of the 60-item scale originally developedby Rotter (1966). It is higher when individuals agree with statements that their life isdetermined primarily by fate or factors outside of their control (external control) and lower
when people respond that they direct their own lives (internal control). I use the measureof self-esteem from 1980, mastery from 1992, and locus of control from 1979.
Table 1 displays pairwise correlations among income and the proposed h variables. All
correlations are significant when run as regressions between income and the h variable.
Years of education and AFQT have the strongest correlations with income. Height has
the weakest, though still positive. Higher self-esteem and mastery are predictive of higher
income, and workers with a more external locus of control have lower income.
To reinforce the point that not any measure of skill will be an unbiased proxy for the
Rosen frontier, I also re-run the analysis using several skills measured in the NLSY that are
intended to measure occupation-specific skills rather than those useful in a broad varietyof occupations. For these occupation-specific skills, I turn to the section-specific scores of30

Figure 7: Revisiting Safety: Comparison of h Variables(OLS estimate)

Qualification Measure

Locus of Control
Mastery
AFQT
Self Esteem
Height (rel to gender)Years of Ed
-1
-.8
-.6
-.4
-.2
0
.2
Compensating Differential Paid in Income for 1SD Safety

Notes: The horizontal axis represents the coefficient on the amenity in a regression of incomeon the one amenity controlling for ĥ, the predicted values from a regression of h on income
and the amenity. Negative values indicate that workers on average lose income to acquire theamenity. All regressions are on the NLSY sample and run at the occupation level, weighted
by number of workers. Anderson-Rubin 95% confidence intervals shown at the occupationlevel. Additional sample details are found in Section V.A.
the Armed Services Vocational Aptitude Battery. The ASVAB is a battery of 10 tests thatmeasure skills ranging from word knowledge to automobile repair skills (e.g., “what typeof wrench is pictured?”). The logic against using a variable like automobile repair skills
as the direction by which to normalize the ability channel is that it is likely to contain
information about a worker’s job preferences by way of specialization in job-specific skills.
It thus violates the identification assumption because it is correlated with how workers split
their compensation into wage and amenities.
V.C.2

Results

The main result of this section is presented in Figure 7, which compares estimates for the
price of safety using each of the six proposed h variables. These variables span widelydifferent spaces of physical, cognitive, and social differences that workers bring to the labormarket. Yet, they all lead to very similar magnitudes of price estimates. All six confidence
intervals overlap with each other, and none overlap with the OLS coefficient of income on
safety.

31

Figure 8: Detailed Comparison of Height vs AFQT

nonprofit

sitting
regularschedule
unstructured
repetition

Job Characteristic

safetydecisionfreedom
timepressure
-1

-.5
0
.5
Compensating Differential Paid in Income
Height1

AFQT

Notes: This graph compares the results when the h variable used is AFQT versus heightresidualized by sex. The correlation between residualized height and AFQT in the NLSY
is 0.1504 at the individual level and 0.3974 at the occupation level. The horizontal axis
represents the coefficient on the amenity in a regression of income on the one amenity
controlling for mean h in the NLSY sample at the occupation level. Additional sample
details are found in Section V.A.
In contrast, Figure A.1 additionally includes the 10 component scores of the ASVAB,
and thus plots 16 estimates using the proposed estimator. These tests are developed tomeasure relative skills, rather than the generally ability channel that might proxy for totalcompensation. As predicted, not all of the ASVAB components even yield the same sign,with the ASVAB Auto & Shop score yielding an estimate even more extreme than the OLS
estimate. Such a finding is to be expected given that this skill measures aptitude for amechanic job, which is one of the most dangerous occupations by the O*NET measure. On
the other end of the estimates is “coding speed,” a test which provides a table of words andnumbers and asks which number corresponds to a certain word. Dropped from the ASVAB
in 2002, the score was previously used to classify military recruits into clerical jobs (Heldand Carretta, 2013). Clerical jobs tend to be relatively safe.
Among several skills that are plausibly relevant to a general set of occupations, I obtain
similar price estimates for safety. Such proxies include the six pre-specified h variables, andex-post, even some ASVAB component scores (e.g., math) also yield similar results. On
the other hand, if the ability channel is normalized to a skill that is designed to measure32

aptitude in a very narrowly defined set of occupations (e.g., car repair skills), I obtain a very
different estimate. This is a normalization that the researcher must think critically about.
To examine the extent to which different h variables produce similar results for a wider
range of amenities, Figure 8 zooms in on two leading candidate h variables: height and
AFQT. Estimates for several amenities, priced separately, are presented to investigate
whether the two candidate h variables, which share a correlation of only .1486 at the individual level, produce similar estimates. Although height is much less precise, there is noevidence to indicate that these variables are leading to different estimates. The empiricalanalysis of this section is consistent with my claim that any of several observed h variables
can serve as proxies for the factors that determine a worker’s level of total compensation.

V.D

Multivariate prices

For simplicity, all of the empirics so far have priced amenities against income in a bivariate
nature. However, the estimator readily extends to a multivariate pricing equation by including the additional amenities in both the first stage (generating predicted values) and in thesecond stage (calculating prices). Mechanically, this is equivalent to the ratio-of-coefficients
approach in which h is regressed on all forms of income and amenity compensation. The
negated ratio of each amenity coefficient over the income coefficient is equivalent to thecorresponding price estimate from the predicted values approach.
Figure 9 compares linear price estimates when amenities are priced in isolation as opposed
to in the multivariate framework. In general, the magnitudes of amenity prices shrink when
they are priced jointly. That the multivariate coefficients are less than the bivariate ones
can be seen in the scatterplot of Figure 9 as a slope of the points less than the red 45◦ line.
This finding is consistent with the theory that good job attributes come bundled together.
That is also what would be expected if income and amenities are viewed as two forms ofcompensation firms use to attract more productive workers.

VI

Demographic Gaps

With estimates of the price of various amenities in the US labor market, I next turn to thequestion of quantifying the extent to which these amenities, at current prices, contribute towidely studied demographic pay gaps. This exercise is a decomposition of income inequality
into an absolute component due to differences in offer curves and a relative component due
to workers’ locations across the income-amenity frontier. Before accounting for a wide arrayof amenities, it is instructive to first make the question concrete in the context of just one
amenity.

33

.5

Figure 9: Separate & Multivariate Amenity Prices

Together0

timepressure

unstructured

safetysitting

regularscheduledecisionfreedom
repetition-.5

nonprofit-.50
Separate.5

Notes: The scatterplot compares linear price estimates when amenities are priced separately(on the horizontal axis) versus in a multivariate framework (on the vertical axis). The hvariable is AFQT. The red line is the 45◦ line. Additional sample details are found in Section
V.A.

VI.A

Graphical Intuition: Regular Schedules

To begin with some graphical intuition, I focus on a variable describing how regular the
job’s schedule is. Although safety seemed to be the quintessential job amenity in the daysof Adam Smith, this variable seems to more closely mirror modern scholarship surrounding
amenities in the gender gap.
The NLSY asks workers whether they “usually work a regular daytime schedule or someother schedule” at their current job. Of those who do not work a regular schedule, themost common alternatives are either irregular schedules arranged by employers or regularschedules containing evening or night shifts. About 80% of workers report working a regular,
daytime schedule. Significant differences exist by race and gender; women and whites eachwork more regular schedules than their comparison groups. Admittedly, this variable may
not be an ideal measure of the more nuanced job attributes relating to scheduling discussed
in the literature on the gender gap, such as whether the nature of the work entails a convex
return to hours as in Goldin (2014). However, if women tend to have have higher valuation
for a regular daytime schedule – for instance, to be home for children in the evenings – thenit may still be interesting to see how costly this noisy and narrowly defined amenity turns
out to be in the data.
34

For the purposes of graphical exposition, let us assume that the true model is linear, andI am only interested in pricing this one amenity in relation to income. (That is, I will obtain
the price of this amenity given how other unobserved amenities covary.) This price of theamenity was already reported in Figure 8: 1 SD of schedule regularity when AFQT is usedas the h variable costs .29 SD of income.40 This price estimate corresponds to the slopeof the Rosen wage-amenity frontiers. That is, if I observe two workers who have different
bundles of wage and amenity, the slope from the price estimate is enough to discern whether
the workers lie on the same or different frontiers. I can thus decompose income inequality
between men and women into a component that is due to sitting on different Rosen frontiers(differences in total compensation) versus differences in locations along the wage-amenity
frontiers (differences in valuations).
Panel A of Figure 10 shows the results of this analysis by binning the labor market into
deciles of total compensation. There is a clear trend: at any level of compensation, men
on average have higher earnings and less regular schedules than women. The difference issubstantial; averaging over all the bins men earn about .08 SD more than women conditionalon total compensation. To put the number in perspective the overall gender income gap inthis sample is .54 SD. This simple result implies that .08/.54 = 14% of the gender income
gap can be attributed to this trend of costly substitution by women to jobs that, roughly
measured, have regular schedules (and other features that go with it).
Panel B shows the identical analysis, but grouped by race rather than by gender. The
pattern is vastly different. Regardless of the well-known income differences between racial
groups, their mean locations are essentially indistinguishable once total compensation isconditioned on. On average, a black and a white worker who face the same job prospects do
not make meaningfully different choices in the same way that men and women do. At least,
blacks and whites do not meaningfully sort along the dimension measured by how regular
the schedule of the job is.

VI.B

Pay Gap Accounting

To get a more holistic picture of the contribution of job amenities to the demographic pay
gaps, I next consider a multivariate pricing equation of, in addition to regular schedule, thefollowing job characteristics: nonprofit, sitting, unstructured, repetition, decision freedom,safety, and time pressure. These are the same characteristics priced separately and together
in Figure 9. Two of the job characteristics (nonprofit and regular schedule) are not available
in the PSID, so are not included for the analysis of inequality by parent background. For
40 In the previous section, I priced all amenities at the occupation-level because that was the level atwhich many were observed. This amenity is observed at the individual-level. Qualitatively, the result of
this analysis are similar whether the amenity is priced at the occupation- or individual-level, although theprice is smaller and less significant when this amenity is priced at the individual-level. This difference isconsistent with the binary amenity measure containing some measurement error at the individual-level,causing attenuation, which is mitigated when the analysis is aggregated to the occupation level.

35

Figure 10: Valuing Regular Schedules(a) Differences by Gender(b) Differences by Race

Notes: Both panels depict superimposed gradients of slope -.29, corresponding to the linear
price estimate of this amenity when AFQT is used as the h variable; both axes are instandard deviations. The widths of each of the 10 bandwidths are not drawn to scale, butloosely corresponds to the region of the graph in which workers who obtain a particular decileof compensation would fall. The exact bandwidths used for calculations were calculated
in the following way. First, AFQT was regressed on income and regular schedule at theoccupation level weighted by number of workers; then, at the individual level, predictedvalues were calculated. The predicted values were grouped into 12 quantiles, and the top andbottom quantiles were dropped. This left 10 quantiles of predicted values, which correspondto (winsorized) quantiles of total compensation. The data points were constructed in thefollowing way using the exact quantiles. For Panel A, I plot the average income and amenity
values for men and women, separately, by decile of total compensation. Mens’ locations are
summarized by blue dots, and women’s locations by red diamonds. Panel B is constructed in
the exact same way, but separately by race rather than by gender. Red diamonds summarize
the locations of “black” respondents, and blue dots refer to respondents who identify as “nonblack, non-hispanic.” The data underlying the figure is the main NLSY analysis sample;
additional sample details are found in Section V.A.

36

0

$ Income Gap
10,00020,00030,000Figure 11: Income Gaps vs. Gaps in Frontiers

genderparents

Raw Income Gap

race
Gap Controling for Frontiers

Notes: The height of the blue bar is the unadjusted income gap among all employed respondents. Data by gender (men minus women) and race (non-black/non-hispanic minus black)are from the NLSY in 2012, and data by parents (above minus below median parent income)are from PSID in 2013. The red bar is the coefficient on the demographic group in a regression of income on the group with a linear control for the multi-dimensional Rosen frontier.
The frontier prices the following amenities: nonprofit (NLSY only), sitting, unstructured,
repetition, decision freedom, regular schedule (NLSY only), safety, and time pressure. In
the NLSY, AFQT is used as the h variable; for the PSID results by parent background, theh variable is education. Additional sample details are found in Section V.A.
the analysis in the NLSY, I use AFQT as the h variable; as this variable is not available in
the PSID, I use years of education to price amenities among that group. I re-price amenities
in the PSID because after restricting the PSID to respondents for whom I observe parental
earnings, I end up with a much younger sample than that of the NLSY; the samples thus
may not be representative of each other.
Effectively, the many dimensions of job heterogeneity allow me to construct multidimensional Rosen frontiers. If the result of the previous section was that men tend tosit on the income side of the income-regular schedule frontier, the question I now ask iswhich side of a high-dimensional income-amenity contour men sit on.
As usual, I define income gaps by sex (male minus female) and race (non-black/nonhispanic minus black). To operationalize inequality by parent income in binary terms, I
define an income gap between those in the PSID based on whether one’s parents earned
below or above the in-sample median. Figure 11 displays the results of this analysis, as well37

Table 2: Income Gaps

Male
White
Above-Median Parents
Constant
Frontier Control
N(1)29296.4(2)
19396.041250.9
4868

(3)

(4)

22407.1-1959.1

41992.6
X
4868

3977

X
3977

(5)

(6)

21813.6
39988.8

1409.2

929

X
929

Notes: All columns report linear regressions in which the dependent variable is income in2012 (or 2013, for the parental sample in columns 5 and 6). The sample is the main NLSY
analysis sample with no hours restrictions for columns 1-4, and the PSID analysis sample
for columns 5 and 6. Additional sample details are found in Section V.A. Standard errors
are not included for this table due to the difficulty of calculating them for the specifications
controlling for the frontier. The controls for Rosen frontiers in columns 2, 4, and 6 areconstructed as described in Section VI.B, in which the frontier prices the following amenities:
nonprofit (NLSY only), sitting, unstructured, repetition, decision freedom, regular schedule(NLSY only), safety, and time pressure. In the NLSY, AFQT is used as the h variable; for
the PSID results by parent background, the h variable is education.
as the overall demographic gaps in the relevant sample. I first start in the main analysis
sample, which has no full-time hours restriction, and then examine robustness of the results
to alternative specifications. As reported in Table 2, in the NLSY analysis sample men
earn on average almost $30,000 more than women, and the race gap is $22,000. The PSID
reports a gap by parent income of $22,000.
Consistent with the graphical intuition of Figure 10, the race gap is not driven bydifferences in income-amenity substitution. Once offer curves are included as controls, there
is essentially no income divide between blacks and whites. Similar can be said of the parent
income gap. Absent a method to calculate proper confidence intervals for the results, a great
deal of caution is warranted in reading too much into the point estimates. Still, if one wereto interpret the directions of the point estimates as signal rather than noise, the story would
be as follows. Although unconditionally blacks earn lower income (which is not shown on
the graph), conditional on an amount of total compensation, blacks also earn slightly moreincome and worse amenities. That is, blacks substitute toward income, and thus the racial
pay gap exists in spite of blacks’ willingness to take on work with worse amenities for higher
pay. The point estimate for the gap by parent income tells the opposite story, in whichkids from poor families are more likely to substitute toward better amenities. Althoughquantitatively small and likely to be imprecise, the point estimate is consistent with thelogic of Jencks and Tach (2005), who noted that if preferences for work characteristics are
passed across generations, some part of the intergenerational income gradient is likely due
to amenity substitution. Across alternative specifications to be discussed in Section VI.D,38

the directions in which amenities point for both the race and parent background gap seems
to be relatively stable.
In contrast to the nature of inequality by race and parent background, which seembetter attributed to general-compensation factors such as skills or discrimination than toincome-amenity substitution, the bulk of the gender gap still exists within observable Rosen
frontiers. These findings are directionally similar to the analysis using only the regular schedule amenity, which found 14% of the gender gap due to the amenity, but are quantitatively
much larger. The analysis with more amenities indicates that approximately two-thirds ofthe gender income gap in this dataset is due to variation along the wage-amenity frontier.
Whether this result is large is a matter of perspective. On the one hand, if the remaining
one-third of the gender gap were due to discrimination in the labor market, this would certainly be cause for concern. But on the other hand, given the limited amount of information
observable about the job, it is possible that if further amenities could be priced, even moreof the gender pay gap might be explained.
At the very least, the findings indicate that the gender pay gap is to a large extent a relative one, whereas those by race and parent background are absolute gaps in total compensation. The absolute differences in compensation may be due to differences in general-purpose
skills such as education, which may themselves be due to discrimination, for instance along
the lines of Neal and Johnson (1996). The absolute differences in job quality may also stemfrom discrimination in the labor market, as in search model of Black (1995). In contrast,
much of the gender gap does not seem to be explained by such general channels. As I outline
next, the gender gap appears to be best explained by the fact that the average man andaverage woman, in addition to having very different incomes, also have observable different jobs in other ways. And differences in job characteristics command prices in the labormarket.

VI.C

Amenity quantities and prices

Figure 12 provides intuition as to how one might decompose the result on the gender pay gap
into prices and quantities of amenities.41 The amenity quantities in Panel A are a simple
but instructive way to think about the implications of this paper for the gender pay gap bydescribing the extent to which the jobs of men and women tend to differ among observabledimensions, without taking a stance on what types of jobs are “better” or “worse.” For
instance, although average income of women is lower than men, the average quantity ofon-the-job safety is .47 SD higher among women.
41 It is worth reiterating that the methodology does not solve issues relating to omitted amenities that are
correlated with observed ones. While the results give credence to the idea that job features are important,the question of which job features matter is inherently subject to which job features the researcher includes.
The goal of this paper is not to argue that a definitive set of job features has been found, but rather topropose a methodology that many researchers can apply with other sets of job features. This exercise should
be viewed as instructive rather than exhaustive.

39

Figure 12: Gender Pay Gap in Quantities and Prices(a) Quantities(b) Prices

nonprofit

timepressure

repetition

unstructured

safetysitting

timepressure

decisionfreedom

sittingsafetyregularschedule

regularschedule

unstructured

repetition

decisionfreedom

nonprofit
-.2

0
.2
.4
Coefficient on Female (Units: Amenity SD's).6-20,000-10,000
0
10,00020,000Multi-Variate Compensating Differential ($'s Per Amenity SD)(c) Quantities x Prices
nonprofit
repetition
safetytimepressure
sitting
regularschedule
unstructured
decisionfreedom-4,000-3,000-2,000-1,0000
Coefficient on Female x Price (Units: $'s)

1,000

Notes: Panel A plots the coefficients from separate regressions of the amenity on an indicator
for female in the NLSY-O*NET sample. The ordering is in accordance with the result inPanel C. To construct Panel B, I first obtained predicted values from a regression of AFQT
on income and all amenities shown at the occupation level, weighted by number of workers.
The bars in Panel B correspond to the coefficients from a regression of income on theseamenities controlling for predicted values. For this figure, yearly income is in dollars rather
than standard deviations. Panel C plots the data in Panels A and B multiplied. Additional
sample details are found in Section V.A.

40

Panel B displays multi-variate price estimates for the same bundle of amenities shownin Panel A. Prices allow us to interpret the differences in quantities. In the example ofsafety, the amenity is valued by the labor market – a positive price, displayed here as anegative compensating differential. Holding fixed other observed job characteristics, workers
on average lose $6,622 of income to obtain 1 SD of safety. A simple back-of-the-envelope
calculation thus suggests, from gender differences in safety alone, one would expect to see a
pay gap of .47 x -$6,621 = -$3,103.42
Panel C systematically applies the same calculation to the rest of the amenities, andis ordered by the product of quantity differences times the price. Across many of theamenities studied, the finding is that gender differences of the amenity at current prices
generate substantial pay gaps in favor of men. Notably, not all amenities included in thisanalysis follow this trend. For instance, men on average tend to have jobs with more freedom
to make decisions, and decision freedom is a job attribute estimated to be costly. However,this countervailing effect is not large.

VI.D

Further robustness checks

The first dimension of robustness I explore in the analysis of demographic pay gaps is the
choice of h variable that proxies for the level of the total compensation frontier. Figure
13 summarizes the main results for the gaps by gender and race for the six candidate hvariables described earlier. I examine only the gaps by gender and race here, as such a wide
set of h variables is not available in the dataset used to measure the gap by parent income.
Between 50% and 75% of the gender pay gap in the main analysis sample persists within
frontiers, depending on choice of h variable. In contrast, the results by race straddle 0 fairly
closely. Reassuringly, the findings on demographic pay gaps are relatively similar regardlessof which h variable is used.
The remaining robustness checks described below are also summarized in Table 3, in theorder in which they are discussed.
Much prior literature has emphasized the sensitivity of the gender pay gap to how one
defines the sample. In particular, differences in hours worked seem to play important roles.
For this reason, Figure A.4 reproduces Figure 11 when the entire analysis is run only on
the sub-sample of workers who report working at least 40 hours per week. As expected, thegender gap in the full-time sample is substantially smaller, at $24,273. However, the gap
controlling for offer curves remains high at $18,049. Thus, amenity substitution accounts
for 74% of the gender pay gap among full-time workers. It is also worth noting that whereas
the gender gap falls considerably upon imposing the hours restriction, the race gap among
42 If the analysis had indicated that jobs with higher safety command a compensating differential of $0,
then I would have been led to conclude that this amenity does not contribute to the gender pay gap. And if
the price were of the opposite sign, then the conclusion would be that the amenity at current prices functions
to close the gender pay gap.

41

Figure 13: Pay Gap Accounting in the NLSY, by h variable(a) Gender Gap(b) Race Gap

Locus of Control

AFQT

Mastery

Mastery

AFQT

Locus of Control

Height (rel to gender)

Height (rel to gender)

Self Esteem

Self Esteem

Years of Ed-20

Years of Ed

0

20
40
60
80
% Gender Gap After Frontier Control

100

-20

0

20
40
60
80
% Racial Gap After Frontier Control

100

Notes: This figure repeats parts of the analysis of Figure 11 with different h variables. PanelA repeats the result by gender, and Panel B repeats the result by race. A similar re-analysis
is not performed for result by parent background because the same set of h variables is notavailable in the PSID. Additional sample details are found in Section V.A.

Table 3: Additional Specifications for Demographic Gaps

GenderRace
Parents

(1)

(2)

(3)

(4)

(5)

(6)

(7)

baseline

full-time

own-sexpricesfemalepricesmale
pricesincludingprestige

individual
prices

.662
-.087
.065

.744
-.052
.068

.815
-.085
.157

.405
-.126
-.001

.756
-.105
.082

.873
-.088
.296

.844
-.056
.213

Notes: All columns report the share of the income gap remaining after Rosen frontiers areincluded as controls, and are described in more detail in Section VI.D. Column 1 reports
these numbers for the specification of Figure 11, and all other columns are perturbationsof this exercise as follows. Column 2 restricts the sample to workers who report at least 40
hours of work per week. Column 3 estimates prices separately by gender and applies therelevant gender’s prices. Columns 4 applies to the whole sample the prices estimated among
only women, and Column 5 applies prices estimated among men. Column 6 additionally
includes the amenity of prestige sourced from the sociology literature, as discussed in Section
VI.D. Column 7 prices amenities without aggregating to the occupation level. Additional
sample details are found in Section V.A.

42

full-time workers is substantially larger, although amenity substitution still plays little-to-norole for race.
Another concern particularly relevant to the gender pay gap is whether discriminationin the labor market may lead men and women to face different prices. Such a case seems
unlikely if labor market discrimination functions in an absolute way to reduce women’s
total compensation. On the other hand, it may be that discrimination functions in a more
relative way, in which different genders must pay a different wage cost to enter certain typesof occupations. In such a model, the Rosen frontiers for men and women could have different
slopes. To explore this possibility, I calculate amenity prices separately by gender, and use
sex-specific price estimates to build Rosen frontiers. Within frontiers of sex-specific slopes,
82% of the gender gap persists due to amenity substitution. I also perform additional
robustness applying female-specific and male-specific prices to the entire sample, whichyield respective estimates of 41% and 76% of the gender gap remaining. Qualitatively, theseresults all show a large role for compensating differentials in causing the gender pay gap,
regardless of which gender I estimate prices on.
More generally, a major caveat worth reiterating is that the results are likely to besensitive to which amenities are included, as discussed in the Introduction. I do not havea systematic solution to this problem. However, in an attempt to broaden the scope of theamenities beyond the types of measures that have typically been examined by economists,
I turned to the sociological literature, in which several studies relating to occupationalprestige have been conducted dating back to Duncan’s (1961) socioeconomic index and Blau
and Duncan’s (1967) application of the index to occupational mobility across generations.
A key input into the original prestige rankings was surveyed opinion scores. Davis et al.
(1992) provide survey measures of prestige rankings from the 1989 General Social Survey,
but updated for the 1980 Census occupation titles. Occupational prestige was measured
among a national sample of 1,500 respondents asked to rank each occupation’s prestige on ascale from 1 to 9. Not every respondent had to rank every occupation, but sufficient overlap
was included to normalize respondents’ scores. Figure A.5 modifies the analysis of Figure
11 to include the occupational prestige score as an amenity. In this case, the additional
dimension of heterogeneity has magnified each result: 87% of the gender gap and 30% of
the gap by parent income are now attributed to costly amenity substitution, and the race
gap is 8% lower due to this channel.
Finally, although the main analysis has priced amenities at the occupation level, thefact that some amenities are observed from the NLSY at the worker level (regular scheduleand non-profit) begs the question of whether the analysis would be much different if allamenities were just priced at the worker level. In this case, pricing amenities at the worker
level happens magnifies the findings slightly by gender and parent background to 84% and
21% respectively while the race result is still relatively small at -6%. The fact that some43

results are made larger when amenities are priced at the individual level may indicate that
there is meaningful variation to exploit in the relationship between income, amenities, andskills at a more granular level than at the occupation.

VI.E

The naïve analysis

The methodological contribution of this paper is to clarify how to handle simultaneity fromunobserved ability in estimating compensating differentials. Yet the empirical question only
concerns whether substitution between wage and amenities contributes to various pay gaps.
It may be unclear whether unobserved ability is empirically relevant to this question. Thisissue seems particularly relevant to the gender pay gap because observable measures ofability are fairly balanced by gender (or in some cases, such as schooling, tend to favor
women).
A seemingly more straightforward approach to understanding the role of amenities in
the gender income gap would be to simply control for them. If amenities played a large rolein the gender pay gap, which is what my prior results have indicated, perhaps one would
expect to see little-to-no income differences between men and women once statistical controls for amenities are introduced into a regression of income on gender. However, Figure
14 shows that the opposite finding is true. Once my standard set of amenities is included
as a control, income differences between men and women if anything slightly widen. Conventional wisdom might suggest that amenities, as they have been measured here, must notcontribute to the gender pay gap because their inclusion as controls has not reduced theregression coefficient. But that conclusion would be in conflict with my main findings ofthis section.
To evaluate the credibility of the two approaches, one needs to specify what one thinks
is in the structural error term of that regression of income on amenities. In this paper,
I have argued that the unobserved factors that lead workers to earn higher wages – suchas unobserved ability – would also lead to better jobs in other dimensions. If this is thecase, as is shown in Online Appendix D.B, even if the full set of amenities were included,
the regression would still not be “identified,” in the sense that its structural error term
will always be correlated with the regressors. The only case in which this regression would
identify a parameter of interest is if the entire structural error term were also observed andcould be controlled for. In that case, the regression should also have a perfect fit and R2 of
1.

VI.F

Confirming prior results

This paper puts forward a new way to look at typical data on income, amenities, andobserved skills. And yet, the findings of this section are in line with the broader literature.

440$ Income Gap
10,00020,00030,000Figure 14: Income Gaps with Amenity Controls

genderparents

Raw Income Gap

race
Gap w/ Amenity Controls

Notes: This graph mirrors the construction of Figure 2, in which he height of the blue baris the unadjusted income gap among all employed respondents. Whereas the red bars of
Figure 2 represented the income gaps controlling for Rosen frontiers (the ability channel),this graph simply plots the coefficient on the demographic group variable on a regression ofincome on the group controlling for amenities. The sample definition and set of amenities
are eactly the same as Figure 2.
A great deal of literature on gender inequality has increasingly moved toward understanding compensating differentials. A sweeping analysis of the gender pay gap by Goldin(2014) pointed a finger toward pay schedules that are non-linearly increasing in hours, particularly in male-dominated occupations. In a recent review of literature on the gender wage
gap, Blau and Kahn (2017) concluded that in the current labor market, with the gender education gap now reversed, conventional human capital variables explain “little of the genderwage gap” while “gender differences in location in the labor market—distribution by occupation and industry—continued to be important in explaining the gap.” These differences
between where men and women tend sit in the labor market is a “factor long highlighted inresearch on the gender wage gap.” Much recent work using stated and real choices has also
drawn attention to the fact that men and women have different distributions of valuations
for various workplace amenities, which, along with prices, drives my results. Recent work byJuhn and Rubinstein (2022) also highlights the role of job sorting between men and womenin pay gaps. While their analysis treats AFQT as a control for ability rather than as aproxy, their model and results yield additional insights about how the tendency for abstract45

jobs to require more work hours than routine jobs contributes to gender pay differentials
through sorting.
Less work has sought to analyze the effect of amenity substitution in other demographic
pay gaps. A notable exception is work by Jencks and Tach (2005) to systematically review
available evidence on the sources of intergenerational income inequality. Some evidence does
exist on the propensity for kids to take on similar roles in the workplace to their parents –for instance, the intergenerational correlation in hours worked documented by Altonji andDunn (2000) and studies documenting trends in self-employment across generations. Jencks
and Tach (2005) offer an educated guess that at least 10% of the intergenerational income
correlation might be due to the effect of a broad class of values passed across generations,
including hours worked, location, and choice of non-monetary work characteristics. In otherwords, they argue that part of the intergenerational income gap is simply due to the tendency of children from lower-income backgrounds to choose lower-paying jobs with better
amenities. My results are most consistent with this theory. A contrary theory, which I havenot encountered in the literature, is that children from low-income families would substitutetoward higher-paying jobs with worse amenities so as to compensate for lost resources ofparent background. Under such a theory, the intergenerational income gap in income terms
would be even larger if kids from poor backgrounds had the same preferences as those fromricher ones; my results are less consistent with that theory.
Although the findings of this section are not entirely new, my procedures entails a significant deviation from the literature. The mounting scholarship on the role of job amenities
in the gender pay gap stands in discord with a literature focused on estimating the prices ofsuch amenities. The central finding of this literature, to echo the words of Bonhomme andJolivet (2009), appears to be a “pervasive absence of compensating differentials.” But thesefindings demonstrate that when more versatile econometric techniques are used to estimate
compensating differentials, a key finding on the importance of job amenities in the gender
pay gap in the current US labor market can also be obtained from widely available datasets.
The econometric techniques can now be applied to datasets of other countries and otherdecades in order to better understand the role of compensating differentials in other income
inequalities of interest.

VII

Conclusion

In their history of instrumental variables regression, Stock and Trebbi (2003) write about an
influential economic text written prior to the discovery of instrumental variables. Based on
the strong positive correlation between price and quantity of iron, economist Henry Moore
had concluded in 1914 he discovered a “new type” of upward sloping demand curve. Since
that time, the vocabulary and procedure of instrumental variables has allowed economists46

to more precisely separate the structural price-quantity relationships of supply and demand.
Calling a regression of wage on amenities with controls a “compensating differential” is as
crude as calling a regression of price on quantity a “demand curve.” Both approaches neglect
the underlying economic structure of the data. Prices and quantities are each caused bydemand shocks and supply shocks; it is necessary to specify which variation one will isolate.
It is now widely known that a few observed supply shocks would more-than identify the demand curve, but that would not be achieved just by including the supply shifters as controls
when regressing prices on quantities. Similarly, a basic economic model of compensating
differentials would specify that wages and amenities are each caused by some notion of absolute ability and relative valuations. Here, too, I have shown that a single ability shifter
can identify the wage-amenity variation due to preferences, but that is not achieved by thetypical approach of including it as a control in a regression of wage on amenity. Instead,this paper develops a new estimator to fit the empirical context that seemed to already beon the tip of many researchers’ tongues: that of having some rough measures of the ability
channel, but not the full picture. The estimator has a similar motivation to instrumental
variables, but differs in that the variable being “instrumented” for – the ability channel –is unobserved. Additional applications of the estimator likely exist in other fields of economics. A necessary feature of problems for which the estimator is useful seems to be thatthe unobserved variable for which a noisy proxy exists is the only unobserved variable in the
structural equation. This feature of the problem allows the unobserved variable to be written as some weighted combination of observed variables, although it is counter-intuitively acausal component of both.
When I apply the proposed estimator to datasets used for decades to study job amenities,
I find price estimates that, relative to typical observational estimates, are more in line withboth quasi-experimental evidence and common sense. This finding confirms the analysis
of Hwang et al. (1992) that the bias from the unobserved ability channel in the typicalobservational approach was substantial. It also answers their call for new econometricmethods to deal with it. With improved estimates of the prices workers must pay for someof the observable dimensions in which jobs differ, I turn to an analysis of demographic pay
gaps. At the estimated prices, I find that job amenities play a prominent role in income
inequality by gender, and essentially no role in the gaps by race or by parent income. Thisfinding highlights that in the current US labor market, income inequality by gender has very
different roots compared to inequalities by race or by parent background.
From a policy perspective, it is important to note that this paper does not find that thegender gap is “caused by” differing valuations of work by men and women. Differences invaluations alone cannot generate pay gaps unless the amenity has a non-zero price. Rather,
the analysis indicates that it is the combination of differing valuations and corresponding
prices of job amenities that generates a substantial portion of the gender gap. A natural

47

economic approach to reducing the gender gap would be to consider whether the prices ofamenities can be changed. In fact, the government already regulates many aspects of theworkplace, including through OSHA and EEOC. My findings imply that efforts to regulate
cheap provision of amenities that are valued by women could have large potential to reduce
the gender pay gap.
An alternative implication, at least at face value, seems more controversial: If womenand men had similar valuations of certain job attributes, then there would be less income
inequality by gender. A nuance of this conclusion is that it is still open for debate the extent
to which differing valuations of job characteristics stem from preferences or from constraints
outside of the labor market. For instance, women may face constraints relating to childcare.
Men may face social norms and expectations to abstain from parental leave. Individual
workers may be unable to remove such societal constraints.
Finally, the empirical exercises provided in this paper are limited in scope. Viewing these
exercises as a proof in concept, there is a great deal more the new estimation strategy canteach us about compensating differentials. These methods can help researchers understandhow amenity prices have changed around the unprecedented shift to remote work brought onby the COVID-19 pandemic. More research should be undertaken to explore the contributions of amenities to other pay gaps, as well as to the intersections of the identities studied
here. Whereas my analysis has focused on a cohort of older workers, questions remain about
how valuations of job features change over the lifecycle, particularly around life events suchas childbirth. The estimation strategy is simple enough that researchers can test their own
hypotheses about the role of their preferred set of job amenities in income inequality, assome researchers have already begun to do.

Bigger datasets with more individual-levelvariation would yield more precision. Future work can exploit individual-level measures ofjob amenities commonly available, such as paid vacation days, child care, and health insurance. Better measures of amenities, using novel data sources or text-to-data approaches,
also emerge as a key area for progress to be made in this area (Sockin, 2021; Zhu, 2023).
The trade-offs between power and noise when aggregating amenities to the occupation-levelshould be better explored, as well as methods for inference relating to demographic gaps.
Lastly, other proprietary and publicly available datasets can help shed light on how the priceof amenities has changed over time and how prices differ across countries’ labor markets. For
instance, how have compensating differentials for jobs providing flexible hours, childcare,or paid maternity leave changed in the US as more women have entered the labor market?
How did the price of jobs with certain schedules change following France’s adoption of the
35-hour work week in 2000? Such analyses can help reveal the responsiveness of amenity
prices both to the changing composition of the labor pool and to differences in government
regulations.

48

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52

For Online Publication: Appendix to“Job Amenities & Earnings Inequality”
by Alex Bell1

A

Appendix Figures and Tables

Figure A.1: The Price of Safety: Comparison of h Variables, including ASVAB Components

Qualification Measure

ASVAB autoshop(OLS estimate)
ASVAB mechanical
ASVAB electronics
ASVAB science
Locus of Control
ASVAB arithmetic
ASVAB math
Mastery
AFQT
ASVAB wordknowledge
Self Esteem
Height (rel to gender)Years of Ed
ASVAB numerical
ASVAB paragraph
ASVAB coding
-1
-.8
-.6
-.4
-.2
0
.2
Compensating Differential Paid in Income for 1SD Safety

Notes: In both panels, he horizontal axis represents the coefficient on the amenity in a regression ofincome on the one amenity controlling for hˆ , the predicted values from a regression of h on income
and amenity. All regressions are on the NLSY sample and run at the occupation level, weighted bynumber of workers. Anderson-Rubin 95% confidence intervals shown at the occupation level.

2

Figure A.2: Rate of Pay

per hourper year
per weekbi-weekly (every 2 weeks)
per month
per daybi-monthly (twice a month)
per item (piece work)by jobother (specify)
payment in kind
0

10
20
30
% of NLSY79 2012 Analysis Sample

40

Notes: The histogram plots the rate of pay of the 4,899 individuals in the primary NLSY analysis
sample. Asked as part of the yearly Employer Supplement, respondents are prompted “what is theeasiest way for you to report your total earnings before taxes or other deductions: hourly, weekly,
annually, or on some other basis?” and if clarification is requested, the researcher will add that thisinformation is necessary to compare the amount that people earn in different jobs. After responding
to this question, the respondent will also be asked to estimate the amount earned in that timeperiod(per hour, year, etc.). If the respondent is a teacher, a prompt will also be made that earnings
should be reported only over the number of months for which the respondent was paid for that job,which is typically 9 or 10 months rather than per year.

3

Figure A.3: Non-Parametric Safety Estimates by Sex(a) Male Only

incomeincome(b) Female Only

safetysafetyNotes: Both panels are constructed in an identical way to Figure 5, except that they split the
sample on sex.

40

$ Income Gap
10,00020,00030,000Figure A.4: Income Gaps vs. Gaps in Frontiers, Hours ≥ 40

genderparents

Raw Income Gap

race
Gap Controling for Frontiers

Notes: This figure reproduces Figure 11, but restricted to workers who report at least 40

hours of work per week.

50$ Income Gap
10,00020,00030,000Figure A.5: Income Gaps vs. Gaps in Frontiers, with Prestige

genderparents

Raw Income Gap

race
Gap Controling for Frontiers

Notes: This figure reproduces Figure 11, but includes the additional amenity of prestige.

6Figure A.6: Indifference Curves are Tangent to the Equilibrium Frontier
Panel A: Multiple Potential Tangencies

Wage W

Possible tangencies

Amenity Z
Panel B: Non-Equilibrium Outcome

Wage W

Amenity Z
Notes: This figure illustrates that if the frontier of jobs chosen does not lie tangent to eachworker’s indifference curve at the point of the job, then the equilibrium is not stable. Panel
A depicts two possible jobs for a worker-firm combination in the lower right corner. One ofthem would result in a frontier that lies tangent to the nearby worker’s job choice, whereas
the other does not. Panel B illustrates that if the non-tangent job is chosen, then it results
in an off-equilibrium blocking situation, in which the nearby worker and the firm acting off
equilibrium could both be made better off if they were matched.

7

Figure A.7: Balanced Idiosyncrasy

Notes: In this simulation, idiosyncratic measurerment error exists in both w and z, thestandard deviations of which are denoted σν w and σν z . The idiosyncrasies are in proporition
such that they balance each other out and the true price is identified. The top panel plots
the data, and the bottom panel relates the magnitudes of the parameters of the simulationw
σ2
to Equaton 10 in Section G, σν2wz = −β γγ z .
ν8Figure A.8: Unbalanced Idiosyncrasy
Panel A: Added variation in w

Panel B: Added variation in z

Notes: Each panel depicts a simulated data-generating process in which Equation 10 in
Section G is not satisfied. In Panel A, σν2w is too large. In other words, there is toomuch idiosyncrasy in wage determination that is not present for amenity determination.
The parameter estimated by my estimator, shown as a dashed line, is larger than the true
parameter β and so the estimated relative price of the amenity is high. In Panel B, thesimulation is calibrated such that σν2z is too large, resulting in a smaller β̂.

9Figure A.9: Choice Framework

Notes: This figure represents hypothetical choice data for two workers, denoted by colors. The
accepted offer of each worker is boxed (Yij = 1). The red arrow indicates the direction in which Yij
is increasing, as is described in Online Appendix J.

10Table A.1: Construction of Amenity Measuresvariable nameregular schedulesourceNLSY

non-profit

safetyrepetition

decision freedom

sitting
unstructured

time pressure

O*NETquestion
Do you “usually work a regular
daytime schedule or some otherschedule?”
Are you “employed by government, bya private company, or a non-profit
organization” or self-employed?
“How often does this job require
exposure to minor burns, cuts, bites,or stings?”“How important is repeating the same
physical activities (e.g., key entry) or
mental activities (e.g., checking entries
in a ledger) over and over, without
stopping, to performing this job?”“How much decision making freedom,
without supervision, does the job
offer?”“How much does this job requiresitting?”“To what extent is this job structured
for the worker, rather than allowing
the worker to determine tasks,
priorities, and goals?”“How often does this job require theworker to meet strict deadlines?”

11

response defining amenity
measureregular daytime

non-profit

1 - “every day”

“extremely important”“a lot”“continually or almost
continually”“a lot of freedom”“every day”

Table A.2: OLS: Individual- vs. Occupation-Level
OLS
Safety SD’s

.078**-0.026OLS
Outcome: Income SD’s
-0.032-0.027

Proposed

AFQT.014***-0.0017.030***-0.0039.072***-0.0071

Constant-.51***-0.047-1.20***-0.15-2.98***-0.32

Level of
Aggregation

Individual (None)

Occupation

Occupation-.32***-0.062

Notes: Standard errors in parentheses clustered at the occupation level. Sample is 4,899
workers in 397 occupations. Standard errors in the final column calculated by bootstrap.
Results in Column 2 are identical if the outcome is maintained at the individual level, sincenone of the regressors vary within-occupation. * p<0.05, ** p<0.01, *** p<0.001

12

B

Solving the Binary Rosen Model

The following model closely follows the structure of Rosen (1986).
Denote by ηi an index of the human capital of worker i. Higher-skilled workers are
more productive. In a competitive market, ηi can also be thought of as an index of thetotal compensation of the worker: the higher the ηi , the better wage and amenity options
available to the worker.

B.A

Worker Utility

Worker i maximizes utility from monetary wages wi and a single non-monetary amenity
zi according to ui (wi, , zi ). Workers supply a single unit of labor inelastically, but have a
choice in which job to work at. To build intuition in a simple model, consider the amenityof cleanliness: the job is either unclean (zi = 0) or clean (zi = 1). Wages vary according
to the worker’s skill level and choice of amenity. A worker of skill level η makes a discrete
choice to work in the dirty job or clean job available to her, yielding ui (wη0 , 0) or ui (wη1 , 1),
respectively.
Define Ci as the wage premium for the unclean job at which worker i is indifferent
between having the amenity versus not: ui (wη1 , 1) = ui (wη1 + Ci , 0). The premium Ci is thecompensating variation required to move the worker from zi = 1 to zi = 0, and it will bepositive if zi is a “good” amenity. Workers with a stronger affinity for the amenity will havea higher Ci .43 In other words, Ci is the amount of wage compensation required to make theworker indifferent between having the amenity and not.
If the amenity were continuous rather than binary, Ci would have a familiar interpretation in terms of indifference curves. Ci would be called the marginal rate of substitution
δuof the worker, defined as δw
δu . Figure 3 Panel A illustrates the interpretation of Ci in theδzcontinuous case. For now, we continue with the example of a binary amenity so that theillustrative model can be solved in closed form.
In contrast to Ci , define ∆wη = wη0 − wη1 as the actual wage premium supplied in market
η for unclean jobs. In other words, ∆wη is the market compensating differential for a worker
of skill level η.
Workers will choose by comparing Ci to ∆wη . Any worker will choose the dirty job if
the wage premium for the dirty job is sufficiently large, that is if ∆wη > Ci .

B.B

Labor Supply

Ci is a random variable with some distribution across workers, perhaps varying by skill
level. Denote the PDF of Ci in market η by gη (C) and CDF Gη (C). Given a market
compensating differential ∆wη , the number of workers in the unclean job is the number ofworkers for whom the actual premium exceeds the compensating variation they demand, for
43 No assumptions are placed on the joint distribution of η13i and Ci .

whom ∆wη > Ci . Integration over the distribution of C determines the supply of workers
to unclean and clean jobs:
SupplyClean=0
=
η

Z ∆wη
gη (C)dC = Gη (∆wη )0

SupplyClean=1=ηZ ∞gη (C)dC = 1 − Gη (∆wη )∆wη

Rosen (1986) discusses various shapes that the distribution of C may take, and their implications for the elasticity of supply. That discussion is beyond the scope of this paper. In
general, higher-variance distributions of C lead to a more elastic supply to z = 1 jobs.

B.C

Firm Production

Analogous to the worker side, firm j maximizes profit function πj (wj , zj ).44 The impact
on profitability of operating clean rather than dirty varies across firms according to the πjfunction. Also parallel to the worker side, each firm is set up to produce using only a single
level of η (production is not elastic to workers of different skills). Whereas firms care about
the worker’s skill level, firms are indifferent to workers of various Ci .
Define Bj as the wage premium for the unclean job at which firm j’s profits are equal
whether it provides the amenity or not: πj (wη1 , 1) = πj (wη1 + Bj , 0). In other words, Bj is ameasure of the cost to firm j of cleaning up its act, or analogously, the benefit of operating
dirty. Symmetric to the case of the worker, the firm will make its choice by weighing off
this benefit relative to the market wage premium ∆wη , and will choose to operate dirty if∆wη < Bj .
If we were analyzing a continuous amenity, Bj would have an interpretation as the firm’sδπmarginal rate of technical substitution, δw
δπ . This equivalence is illustrated in Figure 3 Panel
δz

A.

B.D

Labor Demand

Similar to the case of the workers, characterize the distribution of Bj in a market indexedby skill level η with PDF fη (B) and CDF Fη (B). Aggregate demand for the two types oflabor are:DemandClean=0=ηZ ∞fη (B)dB = 1 − Fη (∆wη )∆wη

DemandClean=1
=
η

Z ∆wη
fη (B)dB = Fη (∆wη )0

44 In this model, we do not think of the amenity as directly influencing the productivity of the worker. It

is purely another form of compensation that is costly for the firm to provide.

14

B.E

Matching Equilibrium

The competitive equilibrium without search frictions will be characterized by assortative
matching of workers to firms in each market. Workers with large Ci will be matched to firms
with low Bi . In other words, workers who must be paid the most to work in a dirty job will
be found in the firms that have the least benefit to production from operating dirty. In eachmarket, when supply and demand equate, the single wage premium for the amenity will be
the fixed point ∆wη that solves Gη (∆wη ) = 1 − Fη (∆wη ).
To more concretely emphasize the connection of the slope of the frontier to g(C) andf (B), Figures B.1 and B.2 simulate equilibrium outcomes from primitives in the case of abinary and continuous amenity, respectively. B.2 shows that the market-wide price of theamenity is responsive to demand changes for the amenity, and each point along the frontier
represents a tangency of a worker’s indifference curve to a firm’s isoquant.
The primitives of the model are distributions of worker preferences g(C) and firm costs
f (B). In the case of a single binary amenity, Ci and Bj are parameters: willingness topay for a unit of the amenity or the lost profits to the firm of providing the amenity. In
the case of a continuous amenity, more generally each worker would have a schedule ofmarginal willingness-to-pay across potential levels of the amenity, and similarly for firms’
costs. In other words, g(C) and f (B) would be distributions over functions mapping wage
and amenity combinations to utility or profits. To reduce the dimensionality of the problem,
my simulation in Figure B.2 takes C and B to be single-dimensional order statistics for
single-parameter functions.
A great deal of theory suggests that workers of different skill levels may have different
Ci . For instance, the “wealth” effects outlined by Weiss (1976) suggest that as workers
experience diminishing utility of income, non-pay job attributes may become relatively more
important. Some recent advances have been made in mapping the distribution g(C). The
surveys of Wiswall and Zafar (2018) and Maestas et al. (2018) have used stated choice
to trace out distributions across workers of these willingness-to-pay functions for several
amenities. Mas and Pallais (2017) have estimated g(C) for alternative work arrangements
among call-center applicants. Consistent with theory, empirical work has typically founda great deal of heterogeneity across workers in willingness to pay for job amenities. If a
researcher had access to good information on both g(C) and f (B) across the full economy,
it would be possible to align these distributions and predict the marginal price faced byeach worker, thus tracing out the frontier ψ. However, we know little from an empiricalstandpoint about f (B), the cost to firms of providing the amenity.
The goal of this paper is not to estimate the distributions of worker preferences nor firmcosts. Instead, the aim is to directly estimate the shape of the ψ frontier. There is a close
theoretical relationship to research that traces the distribution of g(C) for each worker. Ifu(w, z|Ci ) is a function that maps values of w and z to utility, then the slope of the frontier

15

δu

δπ

δw
at the job taken by the worker is equal to δw
δu . It is also equal to δπ at the firm occupied byδz

δz

the worker. The price each worker faces at the margin is a policy-relevant parameter because
it represents how much wage the worker would have to give up for a marginal change in theamenity. A worker’s income plus the amenity times its marginal price is a measure of thetotal compensation (in dollar units) of the worker.

16Figure B.1: Simulated Equilibrium with a Discrete Amenity
Panel A: Equilibrium01
B (Cost of Providing Amenity)

201
C (Valuation of Amenity)2

Panel B: Shift of Non-Marginal Workers’
Preferences

Panel C: Shift of Marginal Worker’s Preference01
B (Cost of Providing Amenity)

201
C (Valuation of Amenity)201
B (Cost of Providing Amenity)

201
C (Valuation of Amenity)

2

Notes: Each panel depicts firms’ costs of providing a discrete amenity (B) and workers’ willingness to pay
for the amenity (C). The red vertical line in each panel represents the market-clearing price between firms
and workers. Firms to the left of the line provide the amenity to workers on the right of the line. To arriveat this price from the generated distributions, I first matched the workers with the lowest C to the firms
with the highest B. This process is not depicted here, but to illustrate, a worker with a C of around .5 wasmatched with a firm with a B of around 1.5. Intuitively, such a match would not result in the provision of
the amenity, as the firm would find it more profitable to supply the worker with .5 more wage income rather
than expend 1.5 of its profits to offer the amenity. I then checked to see at what point the joint distributionsof B and C cross, such that at that point, firms would be indifferent between supplying the amenity andpaying the additional wage, and after that point, firms would not find the amenity profitable. This point isthe equilibrium price at which supply and demand for the amenity clear, which is 1 for Panel A. Panel B
reflects a very different set of preferences, in which some workers’ willingness to pay for the amenity haveincreased such that the distribution becomes right-skewed but the median is unchanged. Intuition would
suggest that increased demand for the amenity and unchanged supply should drive up the price. However,no price change occurs because the marginal worker’s willingness to pay was not affected. In contrast, PanelC demonstrates a distribution of willingness to pay in which the marginal worker has also increased; this,in turn, leads to a price increase.

17

Figure B.2: Simulated Equilibrium with a Continuous Amenity
Panel A: Equilibrium

Frontier

0

2

0

B

w

10

Preferences and Costs

0

0

1

8

C

z

Panel B: Shift of Workers’ Preferences Toward Amenity

Preferences and Costs

0

2

0

B

w

10

Frontier

0

1
C

0

8
z

Notes: Each panel presents a distribution of worker preferences, firm costs, and the resulting
wage-amenity frontier. For this simulation, the worker utility function is parameterized
Cobb-Douglas as a function of Ci , ui = wi1−Ci ziCi , such that workers with higher Ci can besaid to have a higher affinity for z. Similarly, firms’ profits are parameterized as a linear
function of Bj , πj = −Bj wj − zj , such that firms with higher Bj have higher costs ofproviding the amenity. As in the discrete case, workers and firms are negatively matchedon Ci and Bj . Before discussing how the frontier is constructed, it is worth contrasting thefrontiers across the two panels. In Panel B, as workers have come to value the amenity more,
the frontier has tilted clockwise such that, at the margin, the amenity is more expensive.
For illustration, I have drawn the “kissing equilibrium” of firm production frontiers (red)with worker indifference curves (blue) for two workers: one at the 10th percentile of C andone at the 90th.
18

C

Conditional Independence Footnote

In Section III, I used the conditional independence assumption of (w, z) ⊥ h|η to equate
E[h|w, z, η] with E[h|η]. Below, I show this step more formally.

ZE[h|w, z, η] =Hfh|w,z,η (H)dH
Z

fh,w,z,η (H, W, Z, N )dHdW dZdN
fw,z,η (W, Z, N )Z
fh,w,z|η=N (H, W, Z)fη (N )= H
dHdW dZdN
fw,z,η (W, Z, N )Z
fh|η=N (N )fw,z|η=N (W, Z)fη (N )dHdW dZdN by Cond. Indep. Assumption= H
fw,z,η (W, Z, N )Z
fh|η=N (H)fw,z,η (W, Z, N )= H
dHdW dZdN
fw,z,η (W, Z, N )Z= Hfh|η=N (H)dH=H= E[h|η]19

DComparisons with OLS

In this appendix, I discuss what we can learn from the fact that OLS of wage on gender
does not change much with either the inclusion of occupation fixed effects or occupation-level
amenities. I conclude that we cannot hope to learn very much about the role of amenities
from this regression.

D.A

Comparing Assumptions

I will write the specification with amenities as a structural equation so that we can discuss
potential DGP’s for structural errors:
W agei = αGenderi + βAmenityo + γObservedAbilityi + εi + νi
Definitions:• ε is potential idiosyncratic factors (search frictions, networks, etc.)• ν is potential unobserved ability factors (motivation, etc.)
A tempting interpretation of a large α is the following: The gender gap is not “caused by”
amenities that vary at the occupation level. There are four cases in which OLS has this“causal” interpretation here.
1. No ε and no ν
2. No ε and ν exogenous to amenities
3. ε exogenous to amenities and no ν
4. ε exogenous to amenities and ν exogenous to amenties
It would seem prudent to scratch off #1 and #2 quickly, following the advice of Goldberger(1984) and models discussed by Mortensen (2003). At a minimum, #3 and #4 now require
us to believe that idiosyncratic factors like search frictions or networks lead to workers
getting observably better wages, but not better amenities, which does not seem as plausible
as these factors leading to better jobs.
#3 additionally requires that all of ability is observed, which seems stringent. In contrast,#4 relaxes the need to observe all of ability, although it would require a model in which
ability, though it affects wage, does not actually affect amenities. This would be a serious
rejection of the theory of Rosen (1986).
My model is a different model from all of these: ε and ν cause not only income but alsoamenities. This seems to be the model most consistent with the thinking of Rosen (1986).
Income and amenities should be thought of as being caused by differences in constraints
20

(including ability but also effectively idiosyncratic factors) and preferences. This also seems
to be the model consistent with the literature finding “ability bias.”
It happens that this model itself is in conflict with the OLS approach. If my model is thecorrect DGP, then OLS doesn’t have a causal interpretation. Likewise, if any of models #2#4 were correct, then my results wouldn’t have a causal interpretation. The key difference
between my work and the typical approach is that I assume the fifth model. Like in the caseof Goldberger (1984), we get different results when we make different assumptions about
the structural error, but economic reasoning can help guide which assumptions to make.

D.B

OLS cannot “control” for preferences without skill

Hedonic regressions indicate that women earn less than men, even controlling for observablemeasures of skills. A typical regression used to investigate the skill-adjusted income gap(without amenities) would beÊ[wi |hi , F emalei ] = Φ̂hi + α̂F emalei ,which typically yields a negative coefficient on α̂.
One hypothesis is that the remaining inequality is due to differences in amenities between
men and women. This hypothesis has been tested by adding observable occupation-level
amenities to the hedonic regression equation. This approach is thought to make the types ofjobs more comparable, at least insofar as occupation-level amenities are sufficiently specific.
The regression equation looks like this:Ê ∗ [wi |hi , F emalei ] = Φ̂hi + α̂F emalei + β̂zi ,where for operational simplicity we will consider zi to be only a single amenity.
The offer curve we have developed is wi + βzi = γhi + εi . In the context of the model,
the hypothesis that women are discriminated against in terms of total compensation isequivalent to women of the same measured and unmeasured skill suffering from a persistently
lower budget constraint. To denote this, I introduce a shifter α such that wi + βzi =γhi + εi + αF emalei . In other words, for the same z and η, a woman would necessarily
obtain lower w. We assume no difference in the distribution of ε between men and women.
Under this model of the data-generating process, examining the probability limit of α̂yields a familiar OVB term.
α̂∗ =cov ∗ (w, F emale|z, h)cov ∗ (ε, F emale|z, h)=α+var(F emale|z, h)var(F emale|z, h)

The second term is the OVB term and is necessarily negative. To see why, expand theconditional covariance in the numerator to remove the conditioning on z:

21

cov ∗ (ε, F emale|z, h) = cov ∗ (ε, F emale|h) −cov ∗ (ε, z|h)cov ∗ (F, z|h)var(z|h)The first term is 0 as long as unobserved skills do not vary by gender conditional on
observed skills. But in the second term, we will subtract off a positive quantity wheneverz is a valuable amenity (cov(ε, z|h) > 0 ) toward which women have a particular affinity(cov(F, z|h) < 0). This means α̂ will be downward biased in an additive way. Therefore,
even if there were no discrimination in the labor market (the true α = 0), we would expect
to estimate a negative α̂ just because of preferences. Simply controlling for amenities – evenif we observed all relevant amenities at very granular levels – would not remedy this bias.
The nature of the bias stems directly from the unmeasured idiosyncrasy of the offer curve,
ε.

22

E

Multiple Amenities

Consider the generalized equation for the multi-dimensional frontier with various zk variables
representing either different amenities or higher-order polynomials in each amenity:wi +X

βk zk,i =hi + εikThe identification assumption must hold for each zk :cov(εi , wi )cov(hi , wi )=cov(εi , zk,i )
cov(hi , zk,i )Then β̂k can be represented similarly as before as a ratio of regression coefficients fromE ∗ [hi |wi , zi ] = δˆw wi +

X
k

β̂k =

23

δˆzk
δˆw

δˆzk zk,iF

Non-monotonicity of hSection III assumed E[h|η] was everywhere strictly monotone in η. Since E[h|η] was identified by ĥ, this implied that every quantile of ĥ corresponded to a different quantile ofη.
To convert ĥ to quantiles of η in the absence of global monotonicity, the procedure ismodified as follows. First, quantiles of ĥ are constructed as ordinary. If E[h|η] is nonmonotonic, then it is also the case that the function E[h|w, z] is non-monotonic. Thismeans that when plotted in w-z space, some quantiles of ĥ will be disconnected sets, i.e.,they will contain (w, z) tuples that are separated from each other by another quantile of ĥ.
The problem is now that no matter how small we divide quantiles of ĥ, there is still thepossibility the same quantile of ĥ may correspond to two levels of η; for instance, this would
certainly be the case if η is strictly monotone in w-z space. But all that needs to be done isto modify the quantiles of ĥ such that disjoint sets are coded as separate levels. Every levelof the re-coded ĥ quantiles will now correspond exactly to a different quantile of η.

24

G

A Reduced-Form ModelIn the model of Section II, workers chose wages and amenities according to preferences andsubject to a budget constraint with slope β, the level of which was indexed by η. The only
unobserved component of the model was ε, the unobservable skill component. Although
this model allows for a principled application of economic logic to the labor market, thesimplifications limit the extent of the analysis. For instance, what if wage or amenities
are observed by the researcher with imprecision? Such problems of measurement error
are known to contribute to bias in OLS when affecting regressors. How would additional
idiosyncratic variation in wage or amenity bias the estimator?
Whereas the baseline model allowed for unobserved variation only in the determinationof the offer curve, this section extends the model to incorporate additional unspecified
unobserved variation in both wage and amenity determination. The purpose of this exercise
is to determine under what constraints on these idiosyncratic terms β̂ GM M still retains aninterpretation as a relative price of the amenity.
The simplest statistical framework can be described as two equations governing worker
i’s choice of wage wi and amenity zi . First, each outcome is a function of skill level, ηi ,which increases potential wage by γ w and potential amenity by γ z . Second, each outcome
is affected by a sorting parameter indicating the worker’s place along the frontier, which I
refer to as τi . It is important to note that τi has no natural structural interpretation; inparticular, it does not correspond directly to any parameter of the worker’s utility function.
I introduce this reduced-form parameter to capture the result from the structural modelthat – as a result of heterogeneous valuations of both workers and firms – workers will bescattered along the income-amenity frontier, and their place along that frontier is indexed by
τi . Without loss of generality, the variables τi and ηi are independent and can be assumed
to be mean-zero.45 Equations 5 and 6 describe this statistical model.
wi = γ w ηi + βτi(5)zi = γ z ηi − τi(6)wi + βzi = (γ w + γ z δ)ηi

(7)

Equation 7 multiplies Equation 6 by β and adding it to Equation 5. By subtractingout the τ term, Equation 7 provides the mapping of the general statistical model back tothe simplistic budget-curve model of Section II, which did not rely on the reduced-form
45 The definition that τ and η are independent in no way limits the model to a particular class of utilityiifunctions. It is simply a definition of τi relative to similarly skilled workers. Thus, there is not intended to
be a direct correspondence between τi and any parameter of the utility function.

25

parameter τ .

G.AMeasurement Error (Statistical Noise)

The parameter β still represents the relative price of wage and amenity. In this model, if
we observed η then we could learn β from the data. All unexplained heterogeneity between
workers of similar skill could be attributed to preferences, and their differing choices would
trace out a line of slope β.
However, there may be other sources of heterogeneity in wage and amenities that are also
unobserved. In Section III, I address unobserved skill heterogeneity in the model of Section
II by decomposing ηi into an observed part called hi and an unobserved part called εi . But
a variety of competing models – including discreteness of job choices, optimization frictions,and multi-dimensional skills – suggest that not all unobserved heterogeneity in job choices
can be modeled as unobserved inward-outward shifts of the same-sloping offer curve. To
discuss these likely complications to the simple model, I introduce into the statistical model
idiosyncratic perturbations in a worker’s wage or amenity, rather than common shocks toboth. I denote νiw and νiz as independent idiosyncratic components of wi and zi . Below,
Equations 8 and 9 reproduce Equations 5 and 6 with these additional sources of idiosyncrasy.
wi = γ w ηi + βτi + νiw(8)zi = γ z ηi − τi + νiz(9)With the additional level of unobserved heterogeneity, it is not obvious what can be learned
from the data. In the strict model of Section II, any worker with an average level of measured
skills (h) seen with a high wage and average level of amenities would be interpreted as having
a high level of unobserved skill (ε) combined with a strong preference to translate that skill
into wage (τ ) at the prevailing price, β. With the additional sources of heterogeneity, it’sequally likely this is a worker of average skill and average preferences, who simply received
an inexplicably high wage offer, represented by a high ν w .
In order to learn from data that is thought to have been generated by the general
statistical model of Equations 8 and 9, it is necessary to place an assumption of wellbehavedness on these error terms, given by Equation 10 below. If this well-behaved conditionis satisfied, then the same estimator that consistently estimates the parameter of the modelof Section II will also consistently estimate β from Equation 9.
σν2wγw= −β z
2
σν z
γ

26

(10)

Equation 10 states that the variance of the idiosyncratic term in wage relative to thegeometric mean of variance in wage due to preferences and skills is equal to the analogous
ratio for the amenity. Figure A.7 shows a graphical interpretation of this condition by meansof a simulation. If the condition does not hold, it is still possible to discuss the direction of
the bias. As the variance of ν w gets large, the estimate of β̂ will approach infinity, whereas
the estimate of β̂ GM M tends to 0 as the variance of ν z increases. Figure A.8 depicts thetilting of β̂ relative to β in each of these scenarios, as deviations from the simulation of
Figure A.7.

27

Hβ̂ Linear is equivalent to a ratio of coefficients

The interpretation of the estimator as a ratio of OLS coefficients is particularly useful inadapting the model to multiple amenities.
Consider the linear regression of hon w and z:
E ∗ [hi |wi , zi ] = δˆw wi + δˆz zi .

(11)ˆThe estimator is algebraically equivalent to the ratio of these two coefficients, β̂ GM M = δδˆz .
w

The usefulness of the regression of Equation 11 is somewhat surprising given that theerror term, which is thought to contain unmeasured skill, would be correlated with theregressors. Endogeneity of the error term typically biases coefficients. Although there is
not an obvious definition of bias in this case as Equation 11 does not correspond to any
sensible structural equation, it is true that we expect the coefficients δ̂w and δ̂z to change
depending on the precision with which skill is observed. If we suppose workers are on linear
frontiers of Section II such that wi + βzi = ηi = hi + εi (i.e., wage and amenity coefficients
1
∗
and
= 1+λ
of 1 and β, respectively), then the probability limits of the coefficients are δ̂wβδ̂z∗ = 1+λ, where the constant λ is a scaling factor for the relative variances of h andcov(hi ,wi |zi )i ,zi |wi )ε: λ = cov(hcov(εi ,zi |wi ) = cov(εi ,wi |zi ) (the conditional covariances should be read as linear

projections). That this equality holds such that the λ scaling is the same for both probability
limits hinges on the linearized version of Condition 2 holding:cov ∗ (w, h|η) = cov ∗ (z, h|η) = 0

H.A

(12)

Proof

Consider the linear regression of h on w and z, reproducing Equation 11:
E ∗ [hi |wi , zi ] = δˆw wi + δˆz zi .
Proposition: The proposed estimator is algebraically equivalent to the ratio of theseˆ

two coefficients, β̂ GM M = δδˆz .
w

Proof of Proposition
cov(w,ε|z)cov(z,ε|w)= cov(z,h|w)≡ λ for some number λ.
Lemma 1: If Condition 12 holds, then cov(w,h|z)

Proof of Lemma 1
Condition 12 states that cov(w, h)− cov(w,η)cov(h,η)= 0 and cov(z, h)− cov(z,η)cov(h,η)= 0,var(η)
var(η)
cov(w,η)so cov(w,h)
cov(z,h) = cov(z,η) . Since η = h + ε, a condition similar to that of Condition 12 mustcov(w,η)hold, but replacing h with ε, so we also have cov(w,ε)
cov(z,ε) = cov(z,η) . Equating the two steps28

cov(z,h)and re-arranging, we can write cov(w,h)
cov(w,ε) = cov(z,ε) ≡ K for some number K. By a creative

multiplication with 1, K can be rewrittencov(w,ε)cov(z,w)
var(w)
cov(w,ε)cov(z,w)
cov(z,ε)−var(w)

Kcov(z,ε)−K. Since our identificationand cov(w, ε) = cov(w,η)cov(ε,η), we haveassumption yielded cov(w, h) = cov(w,η)cov(h,η)
var(η)
var(η)
cov(w, h) = Kcov(w, ε) and by similar steps cov(z, h) = Kcov(z, ε). These substitutions
allow us to rewrite K further as∗cov(w,h)cov(z,w)
var(w)
cov(w,ε)cov(z,w)
cov(z,ε)−var(w)

cov(z,h)−. Applying the definition of conditional(z,h|w)
covariance, K = cov
cov ∗ (z,ε|w) , and analogous steps can be followed to show that K is also equal∗∗cov (z,h|w)
cov (w,h|z)
to cov(w,h|z)
cov(w,ε|z) . By combining these two expressions of K, cov ∗ (z,ε|w) = cov ∗ (w,ε|z) , which wewill call attenuation parameter λ.
End Proof of Lemma 1
In the model of Section II, wi + βzi = hi + εi . The probability limits of the coefficientsare derived as follows, where all conditional variances and covariances are linear projections:cov ∗ (h, w|z)var(w|z)cov ∗ (ε, w|z)=1−
var(w|z)cov ∗ (ε, w|z) cov ∗ (h, w|z)=1−
var(w|z) cov ∗ (h, w|z)

∗
δ̂w
=

∗
= 1 − δ̂w
λ
1
=
1+λ

cov ∗ (h, z|w)
var(z|w)
cov ∗ (ε, z|w)=β−var(z|w)
cov ∗ (ε, z|w) cov ∗ (h, z|w)=β−var(z|w) cov ∗ (h, z|w)δ̂z∗ == β − δ̂z∗ λ
=

β
1+λ
ˆ

Thus, the probability limit of δδˆz is β when the model of Section II is applied.
w

In the more general statistical model of Section G, the probability limits of the coefficients can be derived in a similar way. As a function of primitives, the outcome-generating
equations are wi = γ w (hi + εi ) + βτi + νiw and zi = γ z (hi + εi ) − τi + νiz . Solving for h, we
w

z

−βν
have the slightly more complicated frontier h = w+βz−ν
− ε.
γ w +βγ z29

cov ∗ (h, w|z)var(w|z)var(w|z) − σν2w
cov ∗ (ε, w|z)=−var(w|z)(γ w + βγ z )var(w|z)cov ∗ (ε, w|z) cov ∗ (h, w|z)var(w|z) − σν2w−=
var(w|z)(γ w + βγ z )var(w|z) cov ∗ (h, w|z)2var(w|z) − σν w
∗
=
− δ̂w
λ
var(w|z)(γ w + βγ z )var(w|z) − σν2w=
var(w|z)(γ w + βγ z )(1 + λ)

∗
δ̂w
=

cov(h, z|w)
var(z|w)
cov ∗ (ε, z|w)
var(z|w) − σv2z−
=βvar(z|w)(γ w + βγ z )var(z|w)2
cov ∗ (ε, z|w) cov ∗ (h, z|w)
var(z|w) − σvz−=βvar(z|w)(γ w + βγ z )var(z|w) cov ∗ (h, z|w)2
var(z|w) − σvz− δ̂z∗ λ=βvar(z|w)(γ w + βγ z )var(z|w) − σv2z
=βvar(z|w)(γ w + βγ z )(1 + λ)δ̂z∗ =The ratio isβvar(w|z) var(z|w) − σv2z
var(w|z)var(z|w) − var(w|z)σv2z=β2var(z|w) var(w|z) − σν w
var(w|z)var(z|w) − var(z|w)σν2w

In the general statistical model, there are two sets of cases in which the probability
limit of the ratio of coefficients is β. First, in the special case in which σv2w = σv2z = 0, we
have a statistical model isomorphic to that of Section II. Second, when the idiosyncratic
σ2

components are well balanced such that σν2wz = var(w|z)
var(z|w) .
v

End Proof of Proposition

30

I

Bias when ID Condition Fails

To focus on the question of bias relating to the identification assumption, rather than issues
of functional form, I assume that the true data generating process is linear in parameters. In
particular, I will use linear regression functions to approximate the following two conditional
expectation functions:

1. E[w|z, η] is linear in z, with parameter β, such that each Rosen frontier can be written:
w − βz = η(13)

2. E[h|η] is not only monotone but also linear. Denote the linear function by γη, and by
Equation 13, the conditional expectation can be written γ(w − βz). Thus, E[h|w, z]
can be approximated by a linear regression of h on w and z.

The empirical approximation to E[h|w, z] is estimated by predicted values ĥ from a linear
regression of h on w and z:ĥ = Ê ∗ [h|w, z]ˆ∗ˆ∗(h,z|w)(h,w|z)w + covz.
In other words, ĥ = λ̂w + κ̂z = cov
var(w|z)ˆ
var(z|w)ˆĥ =λ̂w + κ̂z===ˆ ∗ (h, z|w)ˆ ∗ (h, w|z)
cov
cov
w+
z
var(w|z)ˆ
var(z|w)
ˆˆ
cov(w,z)ˆcov(h,
ˆw) − cov(h,z)
var(z)
ˆ2ˆ
var(w)
ˆ− cov(w,z)
var(z)w+

ˆcov(w,z)ˆcov(h,
ˆz) − cov(h,w)
var(w)
ˆ2ˆ
var(z)
ˆ− cov(w,z)
var(w)z

cov(h,
ˆw)var(z)ˆ− cov(h,ˆz)cov(w,ˆz)
cov(h,
ˆz)var(w)ˆ− cov(h,ˆw)cov(w,
ˆ
z)
w+
z
2
2
var(w)
ˆvar(z)
ˆ− cov(w,ˆz)
var(z)
ˆvar(w)
ˆ− cov(w,ˆz)

The following equations will be useful to simplify some of the sample variances andcovariances:var(ˆ ĥ) = var(ˆ λ̂w + κ̂z)
= λ̂2 var(w)ˆ+ κ̂2 var(z)ˆ+ 2λ̂κ̂cov(w,
ˆ
z)

31

cov(w,ˆĥ) = cov(w,ˆλ̂w + κ̂z)
= λ̂var(w)ˆ+ κ̂cov(w,ˆz)

cov(z,ˆĥ) = cov(z,ˆλ̂w + κ̂z)
= λ̂cov(w,ˆz) + κ̂var(z)ˆThe parameter β can be estimated from the linear regression (the first assumption of
linearity):Ê ∗ [w|z, ĥ] = β̂z + γ̂ ĥFirst, I will derive a simpler expression for β̂ in the linearized model.
covˆ ∗ (w, z|ĥ)β̂ =var(z|
ˆĥ)
cov(w,
ˆz)var(ˆ ĥ) − cov(w,ˆĥ)cov(z,ˆĥ)=

var(z)ˆvar(ˆ ĥ) − cov(z,ˆĥ)2cov(w,
ˆz)(λ̂2 var(w)ˆ+ κ̂2 var(z)ˆ+ 2λ̂κ̂cov(w,ˆz)) − (λ̂var(w)ˆ+ κ̂cov(w,ˆz))(λ̂cov(w,ˆz) + κ̂var(z))ˆ=

2var(z)(
ˆλ̂2 var(w)ˆ+ κ̂2 var(z)ˆ+ 2λ̂κ̂cov(w,ˆz)) − (λ̂cov(w,ˆz) + κ̂var(z))ˆ
cov(w,
ˆz)(λ̂2 var(w)ˆ+ κ̂2 var(z)ˆ+ 2λ̂κ̂cov(w,ˆz)) − (λ̂2 cov(w,ˆz)var(w)ˆ+ λ̂κ̂var(w)ˆvar(z)ˆ+ λ̂κ̂cov(w,ˆz)2 + κ̂2 cov(w,ˆz)var(z))ˆ=2)
var(z)(
ˆλ̂2 var(w)ˆ+ κ̂2 var(z)ˆ+ 2λ̂κ̂cov(w,ˆz)) − (λ̂2 cov(w,ˆz)2 + 2λ̂κ̂cov(w,ˆz)var(z)ˆ+ κ̂2 var(z)ˆ
cov(w,
ˆz)(λ̂κ̂cov(w,ˆz)) − (λ̂κ̂var(w)ˆvar(z))
ˆ=var(z)(
ˆλ̂2 var(w))ˆ− (λ̂2 cov(w,ˆz)2 )λ̂κ̂cov(w,ˆz)2 − λ̂κ̂var(w)ˆvar(z)
ˆ=λ̂2 var(z)ˆvar(w))
ˆ− λ̂2 cov(w,
ˆ
z)2

= −

κ̂
λ̂

= −

cov(h,
ˆz)var(w)ˆ− cov(h,ˆw)cov(w,ˆz)
cov(h,
ˆw)var(z)ˆ− cov(h,ˆz)cov(w,ˆz)

The above expression for the estimator β̂ writes the estimator as a ratio of coefficients,− λ̂κ̂ , rather than as a two-step estimator. I next proceed to take the probability limit of theestimator, plugging the structural equation for the Rosen frontier.
plim β̂ = −−var(η + βz)cov(h, z) − cov(z, η + βz)cov(h, η + βz)var(z)cov(h, η + βz) − cov(z, η + βz)cov(h, z)[var(η) + β 2 var(z) + 2βcov(z, η)]cov(h, z) − [cov(z, η) + βvar(z)][cov(h, η) + βcov(h, z)]var(z)[cov(h, η) + βcov(h, z)] − [cov(z, η) + βvar(z)]cov(h, z)
Re-arranging to make cancellations apparent among terms multiplied by cov(h, z):32−[var(η) + β 2 var(z) + 2βcov(z, η)]cov(h, z) − [cov(z, η) + βvar(z)]γcov(h, η) − [βcov(z, η) + β 2 var(z)]cov(h, z)]cov(h, η)var(z) + βvar(z)cov(h, z) − [cov(z, η) + βvar(z)]cov(h, z)−−[var(η) + βcov(z, η)]cov(h, z) − [cov(z, η) + βvar(z)]γcov(h, η)γcov(h, η)var(z) − γcov(z, η)cov(h, z)var(η)cov(h, z) + γβcov(z, η)cov(h, z) − cov(z, η)cov(h, η) − βvar(z)cov(h, η)cov(h, η)var(z) − cov(z, η)cov(h, z)

Factoring out γ and re-arranging (including removing neg sign)cov(z, η)cov(h, η) + βvar(z)cov(h, η) − var(η)cov(h, z) − βcov(z, η)cov(h, z)
cov(h, η)var(z) − cov(z, η)cov(h, z)
Now we can see conditional covariances take form

β[var(z)cov(h, η) − cov(z, η)cov(h, z)] − [var(η)cov(h, z) − cov(z, η)cov(h, η)]cov(h, η)var(z) − cov(z, η)cov(h, z)
cov(z,η)cov(h,η)β[cov(h, η) − cov(z,η)cov(h,z)] − γ var(η)]
var(z)
var(z) [cov(h, z) −var(η)

cov(h, η) − cov(z,η)cov(h,z)
var(z)
β−var(η) cov ∗ (h, z|η)var(z) cov ∗ (h, η|z)

If cov ∗ (h, z|η) = 0, then bias is none.
Suppose the identification assumption does not hold. In particular, let our h variable
be not just a proxy for η, but furthermore a positive correlate of z even after conditioning
linearly on η, so at every η, cov(z, h|η) > 0 and cov(w, h|η) < 0. This could be the case,for instance, if h were a skill highly valued in high-amenity jobs or if workers with a highamenity preference sought to increase their h.
Of course the variances are positive, and we should typically think of the denominator∗cov (h, η|z) as being positive, since at any level of the amenity, workers with more h are
expected to have better compensation. Therefore, if it is the case that cov ∗ (z, h|η) > 0
then we should expect β̂ < β. If z is a “costly” amenity, then β will be negative. A further
negative bias will cause the amenity to appear even more costly than it actually is.

33

J

Relation to Framework Using Choice Data

The typical choice-theoretic problem is to identify preferences for wage versus an amenityfrom data on job acceptances and rejections. Assume we have designed the choice experimentsuch that individuals only consider wage w and a single amenity z. The data on choices
may be either hypothetical or real. An individual i makes a discrete choice among jobs
indexed by j. The simplest model would be to assume the individual maximizes a linear
utility function with a match-specific error εij and individual-specific term ηi :uij = αw w + αz z + ηi + εij
A common variant of this set-up is to also subscript the α coefficients by i, as preferences
are thought to vary across workers. The mean preferences could be estimated, or a distribution of preferences. Define Yij as an observed indicator for whether individual i chose job
j.
Issues of functional form arise here due to the fact that the observed Yij is binary, whereas
uij is not. McFadden (1973) shows that the conditional logit specification can consistently
estimate the α parameters when εij is Type 1 Extreme Value. In the interest of clarity, let
us put aside these issues of functional form for now.
The key assumption in the choice framework is a correspondence between observed
choices Yij and the level of unobserved utility uij . In other words, workers have at least
some weak tendency to choose the best job in utility terms. The relationship between Yij
and uij is imperfectly observed due to idiosyncratic factors εij . In the data workers may
mistakenly deviate from their optimum, but critically, when workers don’t choose the bestjob, these errors are idiosyncratic; workers don’t systematically choose the higher-wage orhigher-amenity job. The restriction imposed on Y is essentially the same restriction I impose
on h in Section II. Both Y and h correspond, imprecisely but unbiasedly, to more-preferable
jobs. Figure A.9 illustrates this choice model, and how the relative magnitudes of αw andαz are identified as the ratio of coefficients from a regression of choice on wage and amenity,
similar to my proposed estimator that uses observed skills.
Although the estimating equation is similar to my strategy, the interpretation when using
choices as opposed to observed skills is different in a nuanced but important way. In the
typical theory of compensating differentials reviewed in Section II, the quantities estimatedby either approach correspond to a worker’s marginal valuation of the amenity, which isthe slope of the indifference curve. The only difference is where this approximation is being
taken. In the choice framework, the valuation corresponds specifically to the region of job
choices being offered to the worker in the study. In my framework using actual jobs heldamong a representative group of workers, the estimate corresponds to the valuation of theworker who was marginal in the labor market sorting equilibrium. That worker’s marginal

34

valuation is therefore equivalent to the amenity price at that point, which is also the local
slope of the Rosen frontier.

K

Relation to Goldberger (1984)

Goldberger (1984) studied the question of whether race determined wages, above andbeyond qualifications (wage discrimination). A wave of research had recently proposed a
proxy variable approach to overcome bias from mis-measured qualifications. Similar to
my paper, the proxy approach used qualifications such as education as the outcome in aregression on wage and race.46 Goldberger’s insight was that the proxy approach fails if
wage determination is stochastic, denoted by ε in the below equation, where o and u denoteobserved and unobserved qualifications, respectively:
W age = αRace + Qualso + Qualsu + ε
Given data on only wage, race, and Qualso , two mutually independent latent variables still
remain.47 Although Qualso might be a good proxy for Qualsu , it could not be a proxy for
Qualsu + ε. As labor economics has increasingly viewed wage determination as stochastic,
particularly along the lines of search frictions described in Mortensen (2003), the proxyapproach has seemed increasingly ill-suited for the study of wage or salary discrimination.
Whereas the seminal arguments of Goldberger (1984) still ring true, the topic of thispaper differs in that my goal is to control for the only latent variable in a different structural
equation. If wage and amenities are determined by total compensation and preferences, thenwage should be written as a deterministic function of amenities and total compensation.48
The only latent variable is total compensation. Importantly, the proxy approach allows me
to be agnostic about whether it is latent due to unobserved qualifications or ε-type search
frictions leading to better or worse compensation. In this case, it bears emphasis that it is
the typical non-proxy approach that is not robust to the unobserved search frictions affecting
total compensation that were the crux of Goldberger’s critique.

46 Although at any level of education blacks on average had lower wages than whites (the “forward”
regression), a debate surfaced over how to interpret the fact that at any level of wage, blacks averaged lesseducation than whites (the “reverse” or proxy approach). This fact is also true of the modern labor market,
and several authors argued that it implied discrimination against whites (e.g., Hashimoto and Kochin(1980); Conway and Roberts (1983); Kamalich and Polachek (1982)). Regressions of education on wage andrace had been referred to as “reverse” regressions because wage – the outcome of the structural (“forward”)
equation – was moved to the right-hand side. For my application, I do not use the words “forward” and“reverse” because I view wage and amenities each as structural outcomes. That terminology would imply
multiple “forward” equations but no clear “reverse” one relating wage and amenities.
47 Qualsu and εmust be to some degree independent because unobserved qualifications are thought to varyby race, whereas ε (search frictions) do not. This presentation differs slightly from the primary presentationin Goldberger (1984), which replaced Qualsu with measurement error in qualifications, but the logic is thesame. In that case, measurement error and ε should still be thought of as distinct variables because ε isrelated to (causes) wage but measurement error does not.
48 The role of measurement error in wage and amenities is discussed in Section IV.C.

35