Occupational segregation in a Roy model with composition preferences
OOCCUPATIONAL SEGREGATION IN A ROY MODELWITH COMPOSITION PREFERENCES
MARC HENRY AND IVAN SIDOROV
Abstract.
We propose a model of labor market sector self-selection that combinescomparative advantage, as in the Roy model, and sector composition preference.Two groups choose between two sectors based on heterogeneous potential incomesand group compositions in each sector. Potential incomes incorporate group specifichuman capital accumulation and wage discrimination. Composition preferences areinterpreted as reflecting group specific amenity preferences as well as homophily andaversion to minority status. We show that occupational segregation is amplified bythe composition preferences and we highlight a resulting tension between redistri-bution and diversity. The model also exhibits tipping from extreme compositionsto more balanced ones. Tipping occurs when a small nudge, associated with affir-mative action, pushes the system to a very different equilibrium, and when the setof equilibria changes abruptly when a parameter governing the relative importanceof pecuniary and composition preferences crosses a threshold.
Keywords : Roy model, gender composition preferences, occupational segregation, redistribution, tip-ping, partial identification, women in STEM.
JEL subject classification : C31, C34, C62, J24 introduction
Occupational segregation, particularly the under-representation of women in math-ematics intensive (hereafter STEM) fields is well documented (see for instance Kahnand Ginther [2017]). The under-representation of women in STEM fields is an impor-tant driver of the gender wage gap (see Daymont and Andrisani [1984], Zafar [2013]and Sloane et al. [2019], Blau and Kahn [2017]). Insofar as it reflects talent misalloca-tion, it has significant growth implications (see Baumol [1990], Murphy et al. [1991],
The first version is of October 15, 2020. The present version is of December 9, 2020. The authors gratefullyacknowledge helpful discussions with Kalyan Chatterjee, Lena Edlund and Isma¨el Mourifi´e. Correspondanceaddress: Marc Henry, 522 Kern, Department of Economics, Penn State, University Park, PA 16802. E-mail:[email protected]. a r X i v : . [ ec on . T H ] D ec MARC HENRY AND IVAN SIDOROV and Hsieh et al. [2019]). Occupational segregation may also affect welfare directly,through composition effects in utility (see Fine et al. [2020]).Traditional explanations of the self-selection of women out of STEM fields hinge onpecuniary considerations. Differences in labor force participation profiles are blamed.According to Zellner [1975], women weigh early earnings more than career progression.According to Polachek [1981], women minimize the penalty from intermittent work-force participation. Becker [1985] ascribes occupational segregation to differences inproductivity, due to differential household responsibilities. Differences in productivitymay also arise as a result of differences in field specific human capital accumulation asa result of gender stereotyping of learning and occupations, (see Ellison and Swanson[2010] and Carlana [2019]). Finally, incomes of men and women may differ as a resultof wage discrimination in STEM fields (see for instance Buffington et al. [2016]).Beyond pecuniary considerations, selection into STEM may be driven by composi-tion preferences, either directly, through homophily (see Arrow [1998]) or aversion tobeing in a small minority, or indirectly, through amenity provision. Minority stress isdocumented in Meyer [1995]. Sexual harassment is more prevalent in male dominatedoccupations (Gutek and Morasch [1982] and Willness et al. [2007]), and is measuredin terms of compensating differentials in Hersch [2011]. Shan [2020] shows evidence ofincreased attrition of women randomly assigned to male dominated groups in exper-imental settings. Composition preferences may also reflect preferences for amenities,such as workplace flexibility, whose provision depends on gender composition (seeUsui [2008], Glauber [2011], Wiswall and Zafar [2018] and Mas and Pallais [2017]).The model we propose here incorporates both types of forces: monetary and com-position preferences. Individuals from two groups (e.g., women and men) choose theirsector of activity based on their draw from a joint distribution of potential incomes(which incorporate all pecuniary incentives described above) and a preference forcomposition (that reflects group specific amenity provision, homophily and aversionto minority status). Potential incomes are heterogenous within and between groups,whereas composition preferences are heterogeneous between groups only. Our modeltherefore combines the Roy model of self-selection, in Roy [1951], with models where
OY MODEL AND OCCUPATIONAL SEGREGATION 3 behavior is affected by a group average, as in Bernheim [1994] and Brock and Durlauf[2001] or compositions, as in Schelling [1969, 1971, 1973, 1978]. We uncover two types of phenomenon associated with the interaction of compara-tive advantage and composition preference: amplification of occupational segregationand tipping. We show that composition preferences tend to amplify occupationalsegregation relative to compositions induced by comparative advantage only, i.e.,compositions that would arise in a Roy model of sector self-selection without com-position preferences. The scale of the amplification effect depends on the relativeimportance of comparative advantage, on the one hand, and the intensity of compo-sition preferences, on the other. We propose parameterizations of the distribution ofcomparative advantage and composition preference functions that allow us to quan-tify the scale of the amplification effect. We highlight the resulting tension betweenoccupational de-segregation and income redistribution, as the latter increases theimportance of composition preferences relative to comparative advantage. This isone possible mechanism to explain the positive correlation between Gini and femaleSTEM participation in the European Union (see figure 4). Similar forces are foundin Sj¨ogren [1998] and Kumlin [2007].The combination of comparative advantage and composition preferences also pro-duces tipping phenomena. In contrast with Card et al. [2008] and Pan [2015], there isheterogeneity in potential incomes, and tipping occurs from extreme to more balancedcompositions. We document two types of tipping: a shift between two very differentequilibria as a result of a very small nudge (see Heal and Kunreuther [2006]), or asudden change in the set of equilibria, and the disappearance of an equilibrium withextreme compositions as parameters cross a threshold (which corresponds to the no-tion of tipping point in catastrophe theory). We associate the former with quotas (seeBertrand et al. [2019]), and the latter with regulatory changes in workplace culture.As is well known (see Heckman and Honor´e [1990] and references therein), esti-mation of Roy models presents challenges, as the distribution of primitive potentialincomes ( Y , Y ) cannot be recovered from observed realized incomes and sector choice.Mourifi´e et al. [2020] derive sharp bounds on the distribution of potential incomes in a As a mean-field discrete choice game, our model also resembles the Ellison and Fudenberg [2003]model of agglomeration, with two groups but no heterogeneity within group, and Blonski [1999],with binary actions and heterogeneous preferences, but a single group.
MARC HENRY AND IVAN SIDOROV model of sector selection based on incomes only. In the Roy model with compositionpreferences, we derive sharp bounds on the joint distribution of potential incomes andcomposition preference functions based on the observation of realized compositionsand distributions of realized outcomes by sector and by group. Within the paramet-ric specification of comparative advantage and composition preference functions, thisallows us to conduct inference on the relative intensity of composition preferences andpecuniary considerations using existing moment inequality inference procedures.1.
Roy model with composition preferences
Description of the model.
We consider two types of individuals, w and m .There are exogenous masses µ w of type w individuals and µ m of type m individuals.All individuals simultaneously choose between two sectors of activity: sectors 1 and 2.Individuals within each type have heterogeneous potential incomes in each sector,which are exogenously determined. Let ( Y t , Y t ) be the pair of potential incomes ofan individual randomly drawn from the type t sub population, t ∈ { w, m } . Thistype t individual will receive income Y t (resp. Y t ) if they choose sector 1 (resp. 2).This type t individual choses sector 1 (resp. 2) when Y t − h t ( µ t /µ ) is larger (resp.smaller) than Y t − h t ( µ t /µ ), where µ jt /µ j is the ratio of the endogenous mass µ jt of type t individuals employed in sector j over the endogenous total mass µ j ofindividuals employed in sector j , and h t is a non negative non increasing function.Sector 1 advantage ∆ t = Y t − Y t , t ∈ { w, m } has cumulative distribution F ∆ t . It isassumed invertible, with inverse F − t .1.2. Interpretation and structural underpinnings.
The model described in sec-tion 1.1 treats the vector of potential incomes ( Y i , Y i ) of an individual i as a prim-itive. Potential incomes are defined as the incomes individual i would command ineach sector of activity. The potential incomes are heterogeneous at the individuallevel. They incorporate talent, general and sector specific human capital accumula-tion and their price on the labor market. The distributions of potential incomes aretype specific. Different types may have accumulated different amounts of general andsector specific human capital because of differential access to learning, possibly dueto type profiling in society. Different types may also face different prices on the labormarket because of wage discrimination. OY MODEL AND OCCUPATIONAL SEGREGATION 5
The model focuses on the implications of sector choice. Individuals maximize indi-vidual utility, which is assumed quasi-linear, with a non pecuniary component drivenby type compositions. Individuals choose between sectors j ∈ { , } to maximizeutility U ji := Y ji − h t i ( µ jt i /µ j ) , where t i is individual i ’s type. The non pecuniary component in preferences dependsonly on type and sector type compositions, so that, unlike income, there is only typelevel heterogeneity in preferences. The functions h t , t ∈ { w, m } , reflect compositionpreferences. They incorporate pure social interaction effects, where homophily isreflected by non increasing h and aversion to being in a small minority is reflected bylarge values of h ( u ), when u is close to 0. Composition preferences also reflect typespecific preferences for amenities, whose provision depends on composition. Amenitiesinclude flexibility of working hours, physical risk on the job, and workplace culture.1.3. Existence and uniqueness of equilibrium.
A type-sector composition ischaracterized by the vector of masses ( µ w , µ m , µ w , µ m ), where µ jt is the massof type t individuals in sector j , and µ t is the total mass of type t individuals.For t ∈ { w, m } , call r t := µ t /µ t the share of people in subpopulation t who chosesector 1. Since ( µ w , µ m , µ w , µ m ) = ( µ w r w , µ m r m , µ w (1 − r w ) , µ m (1 − r m )) , we seethat ( r w , r m ) characterizes a type-sector composition. In what follows, we will there-fore refer to ( r w , r m ) as a composition.The system is in equilibrium when the type composition in each sector is compatiblewith individual choices. We define it formally as a composition such that no smallpositive mass of individuals has an incentive to jointly switch sectors. For an interiorequilibrium ( r ∗ w , r ∗ m ), i.e., such that ( r ∗ w , r ∗ m ) ∈ (0 , , this is equivalent to the simpleconsistency requirement µ ∗ t /µ t = Pr [ Y t − h t ( µ ∗ t /µ ∗ ) > Y t − h t ( µ ∗ t /µ ∗ )] , for t ∈ { w, m } , where µ ∗ t = µ t r ∗ t , µ ∗ t = µ t (1 − r ∗ t ), µ ∗ j = µ ∗ jw + µ ∗ jm , j ∈ { , } , and where theprobability is computed with respect to the distribution of potential incomes. After Interior equilibrium are Cournot-Nash according to the definition of Mas-Colell [1984].
MARC HENRY AND IVAN SIDOROV some rearrangement, this is equivalent to F − w (1 − r ∗ w ) = g w ( r ∗ w , r ∗ m , /r ) , (1.1) F − m (1 − r ∗ m ) = g m ( r ∗ m , r ∗ w , r ) , (1.2)where ∆ t := Y t − Y t , r ∗ t := µ ∗ t /µ t , r := µ w /µ m , and g t is defined for each ( x, y, z ) , by g t ( x, y, z ) := h t (cid:18)
11 + z yx (cid:19) − h t (cid:32)
11 + z − y − x (cid:33) . (1.3)In the case of a corner equilibrium, the requirement is that the tail of the distribu-tion of sector 1 advantage is not fat enough to overwhelm the disutility of being in avery small minority. For example, ( r ∗ w = 0 , r ∗ m >
0) can only be an equilibrium com-position if for any small mass δ >
0, the tail of type w ’s sector 1 advantage F − w (1 − δ )is smaller than the disutility g w ( δ, r ∗ m , /r ) of being in a small minority. Definition 1 (Equilibrium) . An equilibrium in the Roy model with compositionpreferences is a composition ( r ∗ w , r ∗ m ) with the following properties.(1) If r ∗ w (resp. r ∗ m ) is in (0 , r ∗ w = 0, then there is ¯ δ > < δ < ¯ δ , F − w (1 − δ ) ≤ g w ( δ, r ∗ m , /r ).(3) If r ∗ m = 1, then there is ¯ δ > < δ < ¯ δ , F − m ( δ ) ≥ g m (1 − δ, r ∗ w , r ).(4) Symmetric statements hold for the cases r ∗ w = 1 and r ∗ m = 0.In the case, where F − w (1 − r ∗ ) = F − m (1 − r ∗ ) = 0 , composition ( r ∗ , r ∗ ) is anequilibrium. Indeed, from the definition of g t , we see that g t ( x, y, z ) = 0 in thatcase. In other words, if the median sector 1 advantage is the same for both types,sector type composition may be unaffected by composition preferences. When thefollowing regularity assumptions holds, we also show that existence and uniqueness ofequilibrium obtains if composition preferences do not overwhelm monetary incentives. Assumption 1.
The function h t , t ∈ { w, m } is decreasing and continuously dif-ferentiable. The cumulative distribution F ∆ t of sector 1 advantage ∆ t = Y t − Y t ,t ∈ { w, m } is continuously differentiable with a continuous inverse, and P [∆ t > ∈ (0 , . OY MODEL AND OCCUPATIONAL SEGREGATION 7
Proposition 1.
Under assumption 1, there is ¯ σ > ≤ σ ≤ ¯ σ, thesystem F − w (1 − r w ) = σg w ( r w , r m , /r ) ,F − m (1 − r m ) = σg m ( r m , r w , r ) , has a unique solution ( r w , r m ).When the scale of composition preferences lies below a threshold ¯ σ , there is a uniqueinterior solution. When the scale exceeds ¯ σ , corner solutions appear with extremelevels of occupational segregation.1.4. Amplification of occupational segregation.
Composition preferences mayinduce greater occupational segregation than would result from differences in potentialincome distributions across types w and m . To see this, we first define efficientcompositions, as those sector-type compositions that would result from maximizationof income only. Definition 2.
For each type t ∈ { w, m } , the efficient composition is r et = Pr [∆ t > . Efficient compositions are the compositions that result from income maximization,since individuals with ∆ ≤ >
0) would choose sector 1 (resp. sector 2).Without composition preferences, sector type compositions are determined by primi-tive potential income distributions only. Once we add composition preferences, occu-pational segregation may be amplified, in the sense that type imbalances caused bydifferences in potential incomes may be increased by individual decisions driven byaversion to being in a minority.
Proposition 2 (Amplification effect) . If assumption 1 holds and the efficient com-positions satisfy r ew < r em , then there is an equilibrium ( r ∗ w , r ∗ m ) such that r ∗ w < r ew Figure 1. Illustration of the amplification of occupational segregation. Thebright blue (resp. red) line traces the quantile function of type w (resp.type m ) sector 1 advantage. Fraction r ew (resp. r em ) is the fraction of type w (resp. type m ) individuals in sector 1 in a Roy model with no composi-tion effects. With composition effects, mediated by the g functions definedin (1.3), these fractions move to r ∗ w and r ∗ m . Multiple equilibria and stability. Composition preferences compete withpecuniary incentives in a way that can lead to uneven compositions even when bothtypes have identical primitive potential income distributions and identical preferences.Suppose both types have identical mass µ w = µ m and identical sector 1 advantage,i.e., F ∆ w = F ∆ m := F ∆ . Hence, efficient compositions are also equal, i.e., r ew = r em := r e . Suppose also that both types also have identical composition preferences.In such a configuration, composition ( r e , r e ) is an equilibrium. However, it is notstable if the slope of g is steeper than the slope of F − at that point. Indeed, asmall mass of type w individuals switching from sector 1 to sector 2 would result inincentives for more type w individuals to switch to sector 2 and for type m individualsto switch to sector 1.Figure 2 illustrates this situation, where the parity equilibrium is unstable, whereasthe equilibrium with compositions ( r ∗ w , r ∗ m ) is stable. Despite both groups being iden-tical in preference and potential incomes, and despite no sector being favored a priori,composition preferences push the population to uneven compositions in both sectors. OY MODEL AND OCCUPATIONAL SEGREGATION 9 Figure 2. Illustration of purely endogenous composition imbalances. Bothtypes have identical potential incomes and preferences. The slope of thecomposition preference effect g is steeper at fraction r e than the slope of thequantile function F − . The parity equilibrium composition ( r e , r e ) is unsta-ble, whereas the equilibrium ( r ∗ w , r ∗ m ) with unbalanced type compositions isstable. Occupational segregation, redistribution and tipping effects Parametric model specifications. To analyze the effects of redistribution andaffirmative action in this model, we choose parametric specifications for the distri-butions of sector 1 advantage ∆ t = Y t − Y t , t ∈ { w, m } , and for the compositionpreference functions h w and h m . Assumption 2. There are parameters r et ∈ (0 , , ( c t , C t ) ∈ R and β > such thatthe composition preference function h t and the cumulative distribution function F ∆ t oftype t sector advantage ∆ t = Y t − Y t satisfy, for each u ∈ (0 , and for t ∈ { w, m } , h t ( u ) = c t u and F − t (1 − u ) = C t r et − u [ u (1 − u )] β . The location of the distribution of sector 1 advantage ∆ t = Y t − Y t is governedby r et . The latter is the efficient composition of definition 2, i.e., the proportion oftype t individuals choosing sector 1 in a Roy model without composition effects. The scale of the distribution of sector 1 advantage is governed by C t . Larger values of C t correspond to greater heterogeneity in sector 1 advantage among type t individuals.The preference for composition is hyperbolic, which is intended to model strong aver-sion to being in a small minority. The aversion increases with larger values of theparameter c t . Hence, the ratio c t /C t measures the relative importance of pecuniaryand composition considerations in choice.Parameter β governs the thickness of the tails of the distribution of sector 1 ad-vantage. When 0 < β < 1, the tails of the distribution of sector 1 advantage arenot fat enough to counteract aversion for being in a small minority, so that some fullsegregation of a group can be an equilibrium. If β > 1, the tails of the distribution ofsector 1 advantage are fat enough to counteract aversion for being in a small minority,so that some full segregation is not an equilibrium. The case β = 1 yields closed formsolutions for equilibrium compositions, as detailed in the next subsection.2.2. Occupational segregation and redistribution. Stable equilibrium configu-rations depend on the relative importance of pecuniary and composition considera-tions in choice. We use parameters γ t , t ∈ { w, m } to gauge this relative importance,with γ w := µ m µ w c w C w and γ m := µ w µ m c m C m . (2.1)When both γ w and γ m are sufficiently small, the pecuniary incentive is sufficientlystrong to guarantee existence of equilibrium with interior compositions, i.e., r w and r m both in (0 , γ w or γ m is large enough, composition preferences push the equilib-rium to a corner, where at least one type is confined to a single sector. Proposition 3. Under assumption 2 with β = 1, if 0 < r ew < r em < γ w + γ m < γ := max { γ w r em /r ew + γ m , γ w + γ m (1 − r ew ) / (1 − r em ) } < , the unique equi-librium compositions ( r ∗ w , r ∗ m ) are given by r ∗ w = r ew − γ w − γ w − γ m ( r em − r ew ) ,r ∗ m = r em + γ m − γ w − γ m ( r em − r ew ) . (2.2) OY MODEL AND OCCUPATIONAL SEGREGATION 11 (2) If γ w r em /r ew + γ m ≥ γ m < − r em , the unique equilibrium compositionsare r ∗ w = 0 and r ∗ m = r em / (1 − γ m ).(3) If γ w < r ew and γ w + γ m (1 − r ew ) / (1 − r em ) ≥ 1, the unique equilibrium compo-sitions are r ∗ w = ( r ew − γ w ) / (1 − γ w ) and r ∗ m = 1.(4) If γ w ≥ r ew and γ m ≥ − r em , the unique equilibrium compositions are r ∗ w = 0and r ∗ m = 1.The four equilibrium regions in the ( γ w , γ m ) space are depicted in figure 3. Theamplification effect of proposition 2 is quantified in proposition 3(1). Occupationalsegregation increases with the strength γ t of both types’ composition preferencesrelative to pecuniary incentives. Occupational segregation also increases with theprimitive mean difference r em − r ew between type m and type w sector 1 advantage.A tension arises between wealth redistribution and the reduction of occupationalsegregation. In general, anticipated redistribution reduces sector 1 advantage, hence Figure 3. Illustration of the four equilibrium regions for γ w + γ m < r ∗ w , r ∗ m ). the strength of pecuniary incentives relative to composition preferences. Specifically,a flat tax at rate τ redistributed equally affects sector choice by shrinking sector 1advantage ∆ t to (1 − τ )∆ t . Hence, the effect of redistribution through a flat tax onoccupational segregation is equivalent to a proportional increase of the strength γ t ofcomposition preferences relative to pecuniary incentives to γ t / (1 − τ ). This results inincreased occupational segregation. If ¯ γ < − τ , after redistribution, the equilibriumcompositions ( r τw , r τm ) are r τw = r ew − (1 − τ ) γ w − (1 − τ )( γ w + γ m ) ( r em − r ew ) < r ∗ w ,r τm = r em + (1 − τ ) γ m − (1 − τ )( γ w + γ m ) ( r em − r ew ) > r ∗ m . If, however, ¯ γ ≥ − τ , the effect of redistribution is to push compositions into acorner solution, where at least one type is confined to a single sector. For instance,if γ w r em /r ew + γ m ≥ − τ and γ w ≥ r ew (1 − τ ), type w individuals will all switch tosector 2 as a result of redistribution.The amplification of occupational segregation as a result of redistribution describedin this section is one possible mechanism to explain the positive correlation betweenGini index and the proportion of women among STEM graduates in the EuropeanUnion, seen in figure 4, based on UNESCO’s UIS statistics (STEM shares) and WorldBank indicators (GINI indices).2.3. Tipping and affirmative action. The Roy model with composition prefer-ences displays tipping phenomena of two kinds. One type of tipping effect occurswhen a small deviation in ( r w , r m ) from one equilibrium tips the system to a differ-ent equilibrium. The second type of tipping effect occurs when the set of equilibriachanges abruptly as the parameters ( γ w , γ m ) that govern the relative strength of pe-cuniary and non pecuniary incentives cross a threshold. Unlike residential tipping inthe work of Schelling (1971, 1973, 1978), tipping in this model is beneficial and occursfrom highly segregated states to more even ones. Each type of tipping in this modelis associated with a type of policy. Tipping as a shift between equilibrium is triggeredby policies operating directly on compositions, such as quotas. Tipping as a changein the set of equilibria is triggered by policies operating on preferences or incomes,such as subsidies or changes in amenities, or changes in labor force participation. OY MODEL AND OCCUPATIONAL SEGREGATION 13 Figure 4. Regression of the proportion of women among STEM graduates(UIS data) against the Gini index of countries of the European Union (WorldBank indicators). Quotas and tipping. To analyze tipping between two equilibria, we consider thedynamical system induced by the Roy model with composition preferences. Considertype w individuals for instance. If µ w /µ w > Pr [ Y w − h w ( µ w /µ ) > Y w − h w ( µ w /µ )] , or, equivalently, if r w > F − w (1 − r w ) − g w ( r w , r m , r ), the system is not in equilibrium,and a part of the type w population is induced to move from sector 1 to sector 2.This also affects type m individuals, so that movements of both types of individualshave to be considered jointly. Formally, the dynamical system is given by˙ r w = F − w (1 − r w ) − g w ( r w , r m , r ) , ˙ r m = F − m (1 − r m ) − g w ( r m , r w , /r ) , (2.3)where ˙ r indicates the time derivative of r . Figure 5 is a phase diagram for thisdynamical system in the case of β = 0 . 05 in assumption 2 and specific values for theefficient compositions r ew and r em and the relative strength ( γ w , γ m ) of pecuniary and non pecuniary incentives. Blue curves are the loci of F − w (1 − r w ) = g w ( r w , r m , r )and F − m (1 − r m ) = g w ( r m , r w , /r ). Black arrows indicate the direction of changein ( r w , r m ) off equilibrium. Black (resp. red) dots indicate unstable (resp. stable)equilibria.Holding parameters ( r ew , r em ) and ( γ w , γ m ) fixed, hence holding both sector 1 advan-tage and preferences fixed for both types, a quota of 10% for the representation of eachtype in each sector pushes the dynamical system from any of the corner equilibria tothe unique interior equilibrium ( r ∗ w , r ∗ m ). The latter displays the amplification featureuncovered in proposition 2, i.e., r ∗ w < r ew < r em < r ∗ m . However, given the modestcomposition preferences relative to pecuniary incentives (low values of parameters γ w and γ m ), this amplification effect is small. Figure 5. Phase diagram for dynamical system (2.3) with β = 0 . , γ w = γ m = 0 . r ew = 0 . r em = 0 . 6. There are 17 equilibria, 7 of which arestable (in red), including 6 corner and 1 interior equilibrium. Black arrowsindicate the direction of attraction of the differential system. Tipping occurswhen a small deviation from any of the stable corner equilibria pushes thesystem to the domain of attraction of the interior equilibrium. OY MODEL AND OCCUPATIONAL SEGREGATION 15 Subsidies, change in amenities and tipping. The quota policy entertained insection 2.3.1 operates through a change in compositions ( r w , r m ) while keeping thedistribution F ∆ of sector 1 advantage and preferences h t fixed, t ∈ { w, m } . In contrast,subsidies operate on F ∆ , and amenity changes operate on composition preferences h t without affecting compositions directly. Consider a class of policies that reduceparameters γ w and γ m . They can provoke tipping through the disappearance of a contrarian equilibrium, i.e., a corner equilibrium which excludes from a given sectorthe type that enjoys a comparative advantage in that sector .We describe an example of contrarian equilibrium in the case of β = 1 in assump-tion 2. Let r ew > r em so that type w has an advantage in sector 1. Suppose alsothat γ w r em > (1 − γ m ) r ew and γ m < − r em , so that type w has stronger compositionpreferences than type m . Then, ( r ∗ w = 0 , r ∗ m = r em / (1 − γ m )) is a stable equilibrium.Indeed, lim r w ↓ F − w (1 − r w ) g w ( r w , r ∗ m , r ) = 1 − γ m γ w r ew r em ≤ , so no small mass of type w individuals has an incentive to move to sector 1, nor woulda small deviation create such an incentive. Moreover, F − m (1 − r ∗ m ) = g m ( r ∗ m , , /r ) sothat no small mass of type m individuals have an incentive to switch. Finally, smalldeviations do not create incentives for type m individuals to switch since dF − m (1 − r ∗ m ) /dr m < ∂g m ( r ∗ m , , /r ) /∂r m . Tipping occurs when parameters ( γ w , γ m ) decreasebeyond the threshold defined by γ w r em = (1 − γ m ) r ew . When γ w r em < (1 − γ m ) r ew , thecontrarian equilibrium disappears.2.3.3. Rise in labor force participation and tipping. The model also exhibits anothertype of tipping, through the disappearance of a weakly contrarian interior equilib-rium. In figure 6, a rise in type w labor participation leads to the disappearance of anequilibrium. We call this equilibrium weakly contrarian because type w has compar-ative advantage in sector 1, i.e., ( r ew > r em ), but the equilibrium is such that r ∗ w < r ∗ m .When the weakly contrarian equilibrium disappears as type w labor force participa-tion increases, the only remaining equilibrium is an interior equilibrium that reflectstype w ’s comparative advantage in sector 1, i.e., such that r ∗ w > r ∗ m . This provides Policies that change workplace culture, such as sexual harassment rules, are part of that class. Suppose, for instance, that women have a comparative advantage in finance, but are kept away bya distaste for the socializing practices in that sector (think of visits to strip clubs). an alternative mechanism for the rapid shifts in gender compositions for occupationssuch as bankers and insurance agents documented in Pan [2015].3. Empirical content of the Roy model with composition effects Empirical model. We propose the following empirical version of the Roy modelwith composition preferences. Individual i with type t i ∈ { w, m } chooses sector 1,denoted by D i = 1 if ∆ i > g t i + ξ i , where ∆ i = Y i − Y i , g t i := g w ( r w , r m , r )if t i = w , and g t i := g m ( r m , r w , /r ) if t i = m , and ξ i is a random variable withcumulative distribution function F ξ , independent of ∆ i . Individual i chooses sector 2otherwise, denoted by D i = 2. We interpret this choice model as the result of utilitymaximization as in section 1.2, where the utility includes an additive sector specificrandom utility term. Finally let w ≥ w .Individual i ’s realized income is Y i = Y i when D i = 1 and Y i when D i = 2. Weassume that the four distributions of realized incomes for each type and each sectorare observed in the population. Hence, the quantities Pr( Y i ≤ y, D i = d, t i = t ) are Figure 6. Phase diagrams for dynamical system (2.3) with β = 2 , c w = c m = 3 . , C w = C m = 1 , µ m = 1 , r ew = 0 . r em = 0 . 5. The two diagramsare differ in µ w . On the left diagram µ w = 0 . 8, on the right diagram µ w = 1.So an increase of type w labor force participation leads to the disappearanceof a weakly contrarian equilibrium. OY MODEL AND OCCUPATIONAL SEGREGATION 17 assumed known for all t ∈ { w, m } , d ∈ { , } , and y ≥ w . Type sector composi-tions ( r ∗ w , r ∗ m ) are observed. They are assumed to be equilibrium compositions. Theratio r of total populations of both types is also observed. The cumulative distribu-tion function F ξ of the random utility term is assumed known. The minimum wage w is also assumed known.3.2. Sharp bounds on the primitives. The distribution of potential incomes isnot observed and cannot be directly estimated, since income is only observed in thechosen sector. Therefore, the vector of parameters is not point identified in general .We provide a sharp characterization of the identified set, i.e., the set of parametervalues that cannot be rejected on the basis of the decision model and the observedchoices and realized incomes.Start with simple implications of the model. Take individual i with type t i ,potential incomes ( Y i , Y i ), sector 1 advantage ∆ i , and realized income Y i . Sup-pose individual i is in sector 1 and has a realized income that is smaller than y ,for some y ≥ w . Individual i ’s sector advantage ∆ i = Y i − Y i must then besmaller than y − w , since Y i = Y i ≤ y and Y i ≥ w . Moreover, by the choicemodel, D i = 1 also implies ∆ i > g ∗ t i + ξ i , where g ∗ t i := g w ( r ∗ w , r ∗ m , r ) if t i = w and g ∗ t i := g m ( r ∗ m , r ∗ w , /r ) if t i = m . Hence, we obtain, for all y ≥ w , the impli-cation Pr( g ∗ t i + ξ i < ∆ i ≤ y − w, t i = t ) ≥ Pr( Y i ≤ y, D i = 1 , t i = t ), which gives abound on the unknown distribution of individual i ’s sector 1 advantage ∆ i . Since therandom utility component ξ i is independent of ∆ i , and its distribution is known, thebound can be integrated over ξ i to yield (cid:90) Pr( g ∗ t i + ξ < ∆ i ≤ y − w, t i = t ) dF ξ ( ξ ) ≥ Pr( Y i ≤ y, D i = 1 , t i = t ) . (3.1)The same reasoning applies to the case Y i > y and D i = 2 to yield the bound (cid:90) Pr( w − y < ∆ i ≤ g ∗ t i + ξ, t i = t ) dF ξ ( ξ ) ≥ Pr( Y i > y, D i = 0 , t i = t ) , (3.2)for all y ≥ w , t ∈ { w, m } .3.3. Inference on structural parameters and counterfactual analysis. Whenpotential incomes and composition preferences satisfy assumption 2, system (3.1)-(3.2) defines a continuum of moment inequalities. For each y ≥ w , and each value of See Mourifi´e et al. [2020] for a discussion of partial identification issues in the Roy model. the parameter vector ( r ew , r em , c w , c m , C w , C m ), the moment inequalities can be evalu-ated numerically, and confidence regions derived using existing inference methods onmoment inequality models (see for instance Canay and Shaihk [2017] for a survey).Once structural parameters are recovered, we can evaluate the effect of policiesdescribed in sections 2.3.1 and 2.3.2 in the following way. Given the parameter valuesand the observed equilibrium compositions, the phase diagram for the dynamicalsystem (2.3) determines the equilibrium that would result from the imposition of aquota system, holding structural parameters fixed. In the case of a policy that shiftsthe parameters C t (a subsidy) of c t (an exogenous change in workplace culture), thenew parameter values determine a new phase diagram for the dynamical system (2.3),which also determines the resulting equilibrium.4. Discussion We proposed a model combining comparative advantage with composition prefer-ences in labor market sector selection. This allowed us to propose one mechanismto explain high levels of occupational segregation as well as the observed counterin-tuitive negative correlation between inequality and gender occupational segregation.The model could easily be extended to accommodate preference for parity, where thedisutility function increases on either side of parity. This could be interpreted as atradeoff between homophily and marriage market considerations. Group identity (Ak-erlof and Kranton [2000], Bertrand et al. [2015], Bertrand [2011]), operates throughhuman capital accumulation, but also compositions. For instance, a small ratio ofwomen reinforces the gendered perception of an occupation. Mentoring in the careerdevelopment of women operates through potential incomes, as lifetime earnings areaffected, but it could be incorporated in a multi generation version of the model).However, the sharp separation between comparative advantage, whose effect onself-selection is mediated through potential incomes, and nonpecuniary drivers of se-lection, mediated through compositions and social effects, brings with it some majorlimitations. One major limitation of the model is that composition effects on learningare ruled out. Potential incomes (that incorporate human capital accumulation) areunaffected by type-sector compositions. Peer effects are limited to non pecuniary ben-efits and cannot affect future productivity. The model also rules out effects of gendercompositions on the gender wage gap, through realistic channels such as increased OY MODEL AND OCCUPATIONAL SEGREGATION 19 wage discrimination in STEM fields because of low representation of women. Themodel is a full information game, so that gender specific bias in wage expectations isruled out as a factor in under-representation of women in STEM. It is also frictionless,so there is no place for job search strategies, homophily in referral networks, as inFernandez and Sosa [2005], Zelzer [2020] and Buhai and van der Leij [2020]. Appendix A. Proofs of results in the main text We prove proposition 2 first, then proposition 1, which relies on proposition 2, and finally proposition 3.A.1. Proof of proposition 2. First, note that, under assumption 1, the function g t ( x, y, z ) defined in (1.3)for t ∈ { w, m } is decreasing in x and increasing in y and satisfies g t ( x, x, z ) = 0 for all ( x, y, z ) ∈ (0 , . Next,we construct a non increasing sequence ( r ( l ) w ) l ∈ N , and a non decreasing sequence ( r ( l ) m ) l ∈ N , in the followingway. Define r (1) w := r ew and r (1) m := r em . By definition of ( r ew , r em ), we have F − w (1 − r (1) w ) = F − m (1 − r (1) m ) = 0.Moreover, since r (1) w = r ew < r em = r (1) m by assumption, the properties of functions g t , t ∈ { w, m } , notedabove imply that g m ( r (1) m , r (1) w , r ) < < g w ( r (1) w , r (1) m , /r ) . Hence, F ∆ w ( g w ( r (1) w , r (1) m , /r )) > F ∆ w (0) = 1 − r (1) w and F ∆ w ( g w (0 , r (1) m , /r )) ≤ ,F ∆ m ( g m ( r (1) m , r (1) w , r )) < F ∆ m (0) = 1 − r (1) m and F ∆ m ( g m (1 , r (1) w , r )) ≥ . Now assume ( r ( l ) w , r ( l ) m ) are defined in such a way that F ∆ w ( g w ( r ( l ) w , r ( l ) m , /r )) > − r ( l ) w and F ∆ w ( g w (0 , r ( l ) m , /r )) ≤ ,F ∆ m ( g m ( r ( l ) m , r ( l ) w , r )) < − r ( l ) m and F ∆ m ( g m (1 , r ( l ) w , r )) ≥ . Under assumption 1, functions F − t and g t , t ∈ { w, m } , are continuous, so that there exist ( r ( l +1) w , r ( l +1) m )such that 0 ≤ r ( l +1) w < r ( l ) w and r ( l ) m < r ( l +1) m ≤ F − w (1 − r ( l +1) w ) = g w ( r ( l +1) w , r ( l ) m , /r ) and F − m (1 − r ( l +1) m ) = g m ( r ( l +1) m , r ( l ) w , r ) . Calling r ∗ w and r ∗ m the limits of the monotone sequences ( r ( l ) w ) and ( r ( l ) m ) on [0 , r ∗ w , r ∗ m ) such that r ew < r ∗ w < r ∗ m < r em . Then ⇒ g w ( r ∗ w , r ∗ m , /r ) < 0. Since g w ( r w , r m , /r ) is decreasing in r w and r ∗ w < r ∗ m , we therefore have g w ( r ∗ w , r ∗ m , /r ) ≥ g w ( r ∗ m , r ∗ m , /r ).Hence, we find g w ( r ∗ m , r ∗ m , /r ) < 0, which yields a contradiction, since we know that g w ( r ∗ m , r ∗ m , /r ) = 0.A.2. Proof of proposition 1. Existence of a solution to the system F − w (1 − r w ) = σg w ( r w , r m , /r ) ,F − m (1 − r m ) = σg m ( r m , r w , r ) , (A.1)0 MARC HENRY AND IVAN SIDOROVis guaranteed by proposition 2 in the case r ew < r em . In the case r ew = r em , ( r ew , r em ) is a solution. To showuniqueness, we first show that for small enough σ , any solution is close to ( r ew , r em ). Fix ε > 0. There isa δ > | u | < δ impliesmax {| − r ew − F ∆ w ( u ) | , | − r em − F ∆ m ( u ) |} < ε. By continuity of g t , t ∈ { w, m } , there is a σ > σ max { g w ( r w , r m , /r ) , g m ( r m , r w , r ) } < δ . Hence,for such a σ , any solution ( r ∗ w , r ∗ m ) to (A.1) must satisfymax {| r ∗ w − r ew | , | r ∗ m − r em |} < ε. (A.2)Let ( r ∗ m , r ∗ m ) and (˜ r w , ˜ r m ) be two distinct solutions of system (A.1). Then F − w (1 − r ∗ w ) − F − w (1 − ˜ r w ) = σ [ g w ( r ∗ w , r ∗ m , /r ) − g w (˜ r w , r ∗ m , /r ) + g w (˜ r w , r ∗ m , /r ) − g w (˜ r w , ˜ r m , /r )] . (A.3)Now, given (A.2), F − w (1 − r ∗ w ) − F − w (1 − ˜ r w ) ≥ | r ∗ w − ˜ r w | inf | u − r ew | <ε (cid:20) − ddu F − w (1 − u ) (cid:21) , (A.4)and | g w ( r ∗ w , r ∗ m , /r ) − g w (˜ r w , r ∗ m , /r ) | ≤ | r ∗ w − ˜ r w | sup | u − r ew | <ε (cid:20) ∂∂r m g w ( r w , r ∗ m , /r ) (cid:21) , | g w (˜ r w , r ∗ m , /r ) − g w (˜ r w , ˜ r m , /r ) | ≤ | r ∗ m − ˜ r m | sup | u − r em | <ε (cid:20) ∂∂r m g m (˜ r w , r m , r ) (cid:21) . (A.5)Together, (A.3), (A.4) and (A.5) imply | r ∗ w − ˜ r w | ≤ σ ( K | r ∗ w − ˜ r w | + K | r ∗ m − ˜ r m | ) . (A.6)for some K > K > 0. Similar expressions obtain for the second equation in system (A.1) to yield | r ∗ m − ˜ r m | ≤ σ ( K | r ∗ w − ˜ r w | + K | r ∗ m − ˜ r m | ) , (A.7)for some K > K > 0. For σ < min( K − , K − , K − , K − ) / 2, combining (A.6) and (A.7) yields acontradiction.A.3. Proof of proposition 3. We prove proposition 3 by enumerating all types of equilibria that can ariseunder assumption 2 with γ w + γ m < r ∗ w , r ∗ m ) is an interior equilibrium, i.e., ( r ∗ w , r ∗ m ) ∈ (0 , , then it must solve system (1.1)-(1.2). Hence, ( r ∗ w , r ∗ m ) must be given by (2 . r ∗ w , r ∗ m ) to be in (0 , , we musthave ¯ γ < 1. Conversely, if ¯ γ < r ∗ w , r ∗ m ) is given by (2 . , r ∗ m ) to be an equilibrium, with 0 < r ∗ m < 1, then r ∗ m must solve (1.2), which is equivalentto r em − r ∗ m = γ m (0 − r ∗ m ), and be in (0 , r ∗ m = r em / (1 − γ m ) and γ m < − r em .Moreover, type w individuals have no incentive to switch to sector 1, so that γ w r em /r ew + γ m ≥ r ∗ w = 1and ( r ∗ w , , r ∗ w < 1, are shown to be incompatible with γ w + γ m < References G. Akerlof and R. Kranton. Economics and identity. Quarterly Journal of Economics ,115:715–753, 2000.K. Arrow. What has economics to say about racial discrimination? Journal of Eco-nomic Perspectives , 12:91–100, 1998.W. Baumol. Entrepreneurship: productive, unproductive, and destructive. Journalof Political Economy , 98:893–921, 1990.G. Becker. Human capital, effort, and the sexual division of labor. Journal of LaborEconomics , 3:S33–S58, 1985.D. Bernheim. A theory of conformity. Journal of Political Economy , 102:841–877,1994.M. Bertrand. New perspectives on gender. In O. Ashenfelter and D. Card, edi-tors, Handbook of Labor Economics , volume 4b, pages 1543–1590. Elsevier, North-Holland, 2011.M. Bertrand and E. Duflo. Field experiments on discrimination. In E. Duflo andA. Banerjee, editors, Handbook of Field Experiments , chapter 8, pages 309–393.Elsevier: North Holland, 2017.M. Bertrand, E. Kamenica, and J. Pan. Gender identity and relative income withinhouseholds. Quarterly Journal of Economics , 130:571–614, 2015.M. Bertrand, S. Black, S. Jensen, and A. Lleras-Muney. Breaking the glass ceiling?the effect of board quotas on female labour market outcomes in Norway. Review ofEconomic Studies , 86:191–239, 2019.F. Blau and L. Kahn. The gender wage gap: Extent, trends, and explanations. Journal of Economic Literature , 55:789–865, 2017.M. Blonski. Anonymous games with binary actions. Games and Economic Behavior ,28:171–180, 1999.W. Brock and S. Durlauf. Discrete choice with social interactions. Review of EconomicStudies , 68:235–260, 2001.C. Buffington, B. Cerf, C. Jones, and B. Weinberg. STEM training and early careeroutcomes of female and male graduate students: Evidence from UMETRICS datalinked to the 2010 census. American Economic Review, Papers and Proceedings ,106:333–338, 2016. S. Buhai and M. van der Leij. A social network analysis of occupational segregation.unpublished manuscript, 2020.I. Canay and A. Shaihk. Practical and theoretical advances in inference for partiallyidentified models. In B. Honor´e, A. Pakes, M. Piazzesi, and L. Samuelson, editors, Advances in Economics and Econometrics: Eleventh World Congress , volume 2,pages 271–306. Cambridge University Press, 2017.D. Card, A. Mas, and J. Rothstein. Tipping and the dynamics of segregation. QuaterlyJournal of Economics , 123:177–218, 2008.M. Carlana. Implicit stereotypes: Evidence from teachers’ gender bias. QuarterlyJournal of Economics , 134(1163–1224), 2019.S. Cicala, R. Fryer, and J. Spenkuch. Self selection and comparative advantage insocial interactions. Journal of the European Economic Association , 16:983–1020,2018.P. Cortes and J. Pan. Occupation and gender. In S. Averett, L. M. Argys, andS. D. Hoffman, editors, The Oxford Handbook of Women and the Economy . OxfordUniversity Press, 2018.T. N. Daymont and P. J. Andrisani. Job preferences, college major, and the gendergap in earnings. Journal of Human Resources , pages 408–428, 1984.G. Ellison and D. Fudenberg. Knife-edge or plateau: When do market models tip? Quarterly Journal of Economics , 118:1249–1278, 2003.G. Ellison and A. Swanson. The gender gap in secondary school mathematics athigh achievement levels: Evidence from the american mathematics competitions. Journal of Economic Perspectives , 24:109–128, 2010.R. Fernandez and L. Sosa. Gendering the job: Networks and recruitment at a callcenter. American Journal of Sociology , 111:859–904, 2005.C. Fine, V. Sojo, and H. Lawford-Smith. Why does workplace gender diversity mat-ter? justice, organizational benefits, and policy. Social Issues and Policy Review ,14:36–72, 2020.R. Glauber. Limited access: Gender, occupational composition, and flexible workscheduling. The Scociological Quarterly , 52:472–494, 2011.B. Gutek and B. Morasch. Sex-ratios, sex-role spillover, and sexual harassment ofwomen at work. Journal of Social Issues , 38:55–74, 1982. OY MODEL AND OCCUPATIONAL SEGREGATION 23 G. Heal and H. Kunreuther. Supermodularity and tipping. NBER Working PaperNo. 12281, 2006.J. Heckman and B. Honor´e. The empirical content of the Roy model. Econometrica ,58:1121–1149, 1990.J. Hersch. Compensating differentials for sexual harassment. American EconomicReview, Papers and Proceedings , 101:630–634, 2011.C.-T. Hsieh, E. Hurst, C. Jones, and P. Klenow. The allocation of talent and U.S.economic growth. Econometrica , 87:1439–1474, 2019.S. Kahn and D. Ginther. Women and STEM. NBER Working Paper No. 23525, 2017.J. Kumlin. The sex wage gap in japan and sweden: The role of human capital,workplace sex composition, and family responsibility. European Sociological Review ,23:203–221, 2007.A. Mas and A. Pallais. Valuing alternative work arrangements. American EconomicReview , 107:3722–3759, 2017.A. Mas-Colell. On a theorem of schmeidler. Journal of Mathematical Economics , 13:201–206, 1984.I. Meyer. Minority stress and mental health in gay men. Journal of Health and SocialBehavior , 36:38–56, 1995.I. Mourifi´e, M. Henry, and R. M´eango. Sharp bounds and testability of a Roy modelof STEM major choices. Journal of Political Economy , 128:3220–3283, 2020.K. M. Murphy, A. Shleifer, and R. W. Vishny. The allocation of talent: Implicationsfor growth. Quarterly Journal of Economics , 106:503–530, 1991.J. Pan. Gender segregation in occupations: the role of tipping and social interactions. Journal of Labor Economics , 33:365–408, 2015.S. Polachek. Occupational self-selection: a human capital approach to sex differencesin occupational structure. Review of Economics and Statistics , 63:60–69, 1981.A. Roy. Some thoughts on the distribution of earnings. Oxford Economic Papers , 3:135–146, 1951.T. Schelling. Models of segregation. American Economic Review, Papers and Pro-ceedings , 59:488–493, 1969.T. Schelling. Dynamic models of segregation. Journal of Mathematical Sociology , 1:143–186, 1971. T. Schelling. Hockey helmets, concealed weapons, and daylight saving. Journal ofConflict Resolution , 17:381–428, 1973.T. Schelling. Micromotives and macrobehavior . Norton: New York, 1978.X. Shan. Does minority status drive women out of male dominated fields. Unpublishedmanuscript, 2020.A. Sj¨ogren. The effects of redistribution on occupational choice and intergenerationalmobility: does wage equality nail the cobbler to his last. unpublished manuscript,1998.C. Sloane, E. Hurst, and D. Black. A cross-cohort analysis of human capital special-ization and the college gender wage gap. Becker Friedman Institute working papernumber 2019-121, 2019.E. Usui. Job satisfaction and the gender composition of jobs. Economics Letters , 99:23–26, 2008.C. Willness, P. Steel, and K. Lee. A meta-analysis of the antecedents and consequencesof workplace sexual harassment. Personnel Psychology , 60:127–162, 2007.M. Wiswall and B. Zafar. Preference for the workplace, investment in human capital,and gender. Quarterly Journal of Economics , 133:457–507, 2018.B. Zafar. College major choice and the gender gap. Journal of Human Resources , 48(3):545–595, 2013.H. Zellner. The determinants of occupational segregation. In C. Lloyd, editor, Sex,discrimination, and the division of labor . Columbia University Press, 1975.D. Zelzer. Gender homophily in referral networks: Consequences for the medicarephysician earnings gap.