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Stereotypes, Segregation, and the Dynamics of Racial Inequality

Rajiv Sethi

Racial Classification as a Cognitive Act

Race is a “socially constructed mode of human categorization... for which no deeper justification in biological taxonomy is to be had”

Racial categorization is a “cognitive, not a normative activity“ Loury (2002)

“Racial categories are historical products that are often contested”

“Race still wields monumental power as a social category” Cornell and Hartmann (1997)

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Seminal ContributionsSchelling: Do extreme levels of segregation reflect

extreme preferences over neighborhood racial composition?

Arrow: Can negative stereotypes have incentive effects that cause them to become self-fulfilling?

Loury: What conditions must the process of intergenerational wealth transmission satisfy in order for racial inequality to vanish eventually?

Plan of Talk

Inequality and SegregationDoes declining inequality imply greater integration?

Segregation and StereotypesCan greater integration eliminate negative stereotypes?

Stereotypes and CrimeDo stereotypes affect incentives for criminal behavior?

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Inequality and Segregation

(with Rohini Somanathan)

Schelling on Segregation

If preferences are sufficiently “tolerant” … integrated allocations can be stable… but have small basins of attraction… so a few random movements of households… can “tip” the system into a basin of attraction… of a highly segregated equilibrium

Key insight: segregated allocations are very hard to “escape” (relative to integrated allocations)

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Stability of Segregation

“People who have to choose between polarized extremes ... will often choose in a way that reinforces the polarization. Doing so is no evidence that they prefer segregation, only that, if segregation exists and they have to choose between exclusive association, people elect like rather than unlike environments” (Schelling, 1978, p.146).

Changes in Laws, Inequality & Attitudes

Laws:1964 Civil Rights Act 1968 Fair Housing Act

Inequality: Rapid expansion of black middle classConvergence of black-white household incomes

Attitudes: Significant changes in attitudes of whites

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Most Segregated Cities: Dissimilarity Index

80.4282.7182.85Newark80.8584.4488.39Chicago81.8282.2381.61New York82.1682.7883.95Milwaukee84.7287.4887.46Detroit200019901980

Segregation from Whites - New York Metropolitan Area(New York City plus Long Island and Northern Suburbs)

20

30

40

50

60

70

80

90

1960 1970 1980 1990 2000

Source: Lewis Mumford Center, University at Albany

Inde

x va

lue

Blacks

Hispanics

Asians

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Manhattan: Actual and Hypothetical (2000)

68,450$200,000 or more147,989$200,000 or more

62,930$150,000 - $199,999110,881$150,000 - $199,999

60,024$125,000 - $149,99997,381$125,000 - $149,999

56,837$100,000 - $124,99991,829$100,000 - $124,999

52,611$75,000 - $99,99983,746$75,000 - $99,999

49,282$60,000 - $74,99979,039$60,000 - $74,999

47,312$50,000 - $59,99976,459$50,000 - $59,999

44,951$45,000 - $49,99973,016$45,000 - $49,999

44,725$40,000 - $44,99974,186$40,000 - $44,999

43,606$35,000 - $39,99971,768$35,000 - $39,999

42,476$30,000 - $34,99969,934$30,000 - $34,999

41,256$25,000 - $29,99968,130$25,000 - $29,999

39,785$20,000 - $24,99966,611$20,000 - $24,999

38,417$15,000 - $19,99964,422$15,000 - $19,999

37,160$10,000 - $14,99960,972$10,000 - $14,999

34,941Less than $10,00059,334Less than $10,000

Neighborhood IncomeOwn IncomeNeighborhood IncomeOwn Income

BlackWhite (Non-Hispanic)

Neighborhood Income by Own Income and Race:New York PMSA (2000 Census)

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Neighborhood Income by RaceMean Neighborhood Household Income in New York

0

20000

40000

60000

80000

100000

120000

140000

160000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Household Income CategoryWhite Black

Central QuestionsHow do racial income disparities interact with preferences

over neighborhood characteristics to determine patterns of residential location in metropolitan areas?

Do smaller income disparities imply more integration?Is segregation consistent with strongly integrationist

preferences?Does membership of a lower income group (other things

equal) imply lower levels of neighborhood affluence at stable equilibria?

What are the implications of this for the intergenerational transmission of racial inequality?

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Basic Features of the Model

Incomes vary within and between groupsHouseholds care about consumption,

neighborhood affluence, neighborhood racial composition

Preferences may be pro-integrationist No discrimination Households can locate in any neighborhoodRents adjust to clear the housing market

The ModelContinuum of households; differ by income and race Two groups: “black” & “white”Two neighborhoods of equal size; rents uniform within

neighborhoods

Income distributions: overlapping supports; whites wealthier as a group

Preferences: households care about private consumption, neighborhood mean income, racial composition

Allocations induce neighborhood income distributions and racial compositions

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Preferences

Income is spent on rent and private consumptionHouseholds care about

… private consumption… mean neighborhood income… neighborhood racial composition

Preferences are monotonic in the first two… but non-monotonic in the second… and satisfy “single crossing”

Types of Allocations

An allocation is ...

integrated if each neighborhood contains both groups; segregated otherwise

intraracially stratified if there exist threshold income levels for blacks and whites such that all those above the threshold live together

At any IRS allocation, the neighborhood with households above the thresholds is upper-tail

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Marginal Bid Rents

At any given IRS allocationNeighborhood incomes and compositions are fully

determinedThe marginal bid rent for whites is the maximum

rent that the marginal white household is willing to pay to live in upper-tail neighborhood

The marginal bid rent for blacks is analogously defined

Equilibrium

Equilibrium is an allocation ... of households across neighborhoods ... and a profile of neighborhood rents ... such that no household wants to move

Stable equilibria are such that ... small peturbations are self-correcting

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A Numerical Example

Income distributions are uniform … support [0,0.7] for black households… and [0.3,1] for white households

Black share of population is 45% “Ideal” racial composition is 57.5% own

group

Then there are two equilibria, only one of which is stable ...

Equilibria

Notation: threshold incomes yw, ybneighborhood racial compositions β1, β2neighborhood mean incomes µ1 , µ2

rent differential ρ

0.020.540.490.740.160.120.830.110.680.350.000.900.700.36

ρµ2µ1β2β1ybyw

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Marginal Bid Rents

Properties of Segregated Equilibrium

Threshold incomes: 0.36 (white), 0.7 (black)Neighborhood affluence: 0.35 (LT), 0.68 (UT)White Shares: 10% (LT), 100% (UT)UT Neighborhood Rent: 0.11

Key point: Black households with incomes between 0.36 and 0.7 experience lower neighborhood quality than white households with the same income

This holds more generally

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Main Results

1. If the β is sufficiently high, segregation is stable when racial inequality is either very high or very low

2. If preferences are sufficiently tolerant, there exist multiple stable equilibria when racial inequality is sufficiently low

3. If β is sufficiently high, black households experience lower neighborhood affluence than white households conditional on income

Inequality and Segregation

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Implications

Stable integration becomes viable as income disparities lessen

But historical patterns of segregation may trap a city in basin of attraction of segregated equilibrium

Temporary incentives for segregation may give rise to permanent effects

Lower segregation corresponds to greater stratification (Wilson effect)

From cross-sectional perspective, lower racial inequality may not correspond to lower segregation

From historical perspective, the march towards greater integration may be reversed as racial inequality declines (resegregation)

Intergenerational Effects

Lower inequality in the current generation… if it results in greater segregation… can result in greater future inequality

Policies that reduce segregation… can reduce inequality over time

Next: what effects might desegregation have on the prevalence of negative stereotypes?

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Segregation and Stereotypes

(with Shubham Chaudhuri)

Statistical DiscriminationEx Ante identical groups (Arrow, 1973)

... may face different treatment ex post

... and have different population characteristics

... due to the incentive effects of stereotypes

... which can cause them to become self-fulfilling

Group differences arise in equilibrium ... but are not based on intrinsic differences

This is the axiom of anti-essentialism (Loury, 2002)

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Neighborhood EffectsStatistical discrimination implies racial disparities

... in human capital accumulation

If there is significant residential segregation by race ... there will be racial disparities ... in neighborhood skill distributions

Human capital spillovers at the neighborhood level ... then imply racial differences in the costs ... of human capital accumulation

Key Questions

What effect might lower segregation have on the nature and extent of statistical discrimination?

Can integration eliminate negative stereotypes?

If so, does it enhance or reduce welfare?

Under what conditions can integration eliminate stereotypes and raise skill levels and welfare?

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A Baseline ModelTwo groups of workers (1 and 2); population

share of group 1 is βCost of skill acquisition is cDistribution function for costs F(c); same

distribution in each group Worker investments not perfectly observable Firms assign workers to high-skill or low-skill jobs(Normalized) output in low-skill job is 0 High-skill job output is xq > 0 if qualified, xu < 0 if

unqualified; set xq = 1 and xu = -1

Signals Firms observe a noisy signal which may be

positive (P) or negative (N) Prob(signal is P | worker is qualified) is pProb(signal is P | worker is unqualified) is q < pIf si workers in group i become qualifiedPosterior probability P worker is qualified:

θ(si) = psi/(psi + q(1-si))Posterior that N worker is qualified:

ϕ(si) = ((1-p)si)/((1-p)si + (1-q)(1-si))

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Job Assignment

Firms assign workers to high-skill job if and only if expected output is positive

Workers with P signals get assigned if θ(si) > ½ Workers with N signals get assigned if ϕ(si) > ½ Since θ(si) > ϕ(si) and both are increasing:

None get assigned if si is sufficiently lowOnly P workers get assigned if si is intermediateAll get assigned if si is sufficiently high

Wages and Skill Acquistion

Wages equal expected marginal productwp(si) = max{2θ(si)-1,0}wn(si) = max{2ϕ(si)-1,0}

What is the benefit b(si) of skill acquisition to the marginal worker?b(si) = 0 if none get assignedb(si) > 0 and increasing if only P get assignedb(si) < 0 and decreasing if all get assigned

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Costs of Skills Acquisition

Give that si acquire skills…What’s the cost c(si) of skill acquisition to the

marginal worker?Assuming lowest cost workers acquire skills

…cost c(si) is defined by solution to F(c) = si…which is strictly increasing

Assume F(0) > 0, so that c(0) < 0Then some workers will always acquire skills

Asymmetric Equilibria

(sl, sh) is an equilibrium if and only if (sl, sl) and (sh, sh) are both equilibria

Asymmetric equilibria exist if and only if multiple symmetric equilibria exist

Dynamics: dsi/dt = b(si) – c(si)

Stable asymmetric equilibria exist quite generally

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Segregation and Spillovers

At any population state (s1, s2)Mean skill level in economy is βs1 + (1 - β)s2Mean skill level in neighborhood i satisfies:

σi = ds1 + (1-d) (βs1 + (1-β)s2)where d ∈ [0, 1] is an index of segregationWhen d = 1 (complete segregation): σi = siWhen d = 0 (complete integration):

σi = βs1 + (1-β)s2

Neighborhood EffectsDistribution function for costs G(c,σ):

... proportion of individuals with costs below c

... in group with neighborhood skill level

G is increasing in σ (positive spillovers)... identical for both groups (anti-essentialism) ... but σ may vary across groups in equilibrium

Cost of skill acquisition for marginal worker is… c(si,σi), increasing in si and decreasing in σi

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Multiplicity of Equilibria

(sl, sh) is an equilibrium with d = 1 iff (sl, sl) and (sh, sh) are both equilibria for all d ∈ [0,1]

So an ASE exists under complete segregation iff... there are multiple stable symmetric equilibria... at all levels of segregation

However, existence of an ASE when d = 1 does not imply existence of an ASE for all d ∈ [0,1]

Dynamicsdsi/dt = b(si) - c(si,σi)

= b(si) - c(si,dsi + (1-d)(βs i + (1-β) s 2)

Isoclines: set of points in (s 1,s 2) space along which ds1/dt = 0 or ds2/dt = 0

Any intersection of ds1/dt = 0 and ds2/dt = 0 isoclines... is a rest point of the dynamics... and an equilibrium of the model

But not all equilibria are stable

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Effects of IntegrationStarting from an asymmetric equilibrium (sl, sh)A decline in d raises s1 and lowers s2 at the original

skill shares If qualitative properties of system remain intact, s1

is higher and s2 lower at new equilibrium Increasing integration, if it allows stereotypes to

persist, raises skill levels in disadvantaged group, lowers them in advantaged group

Small increases in integration benefit one group at the expense of the other

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Threshold EffectsThere may exist a level integration above which

stereotypes are inconsistent with equilibrium As integration proceeds beyond this bifurcation

value, there is qualitative change in equilibrium properties of the model

A small change in integration can result in large and discontinuous shift in equilibrium skill levels

Economy converges to state in which groups receive equal treatment

But which such state is reached?

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0 10

1

Skill Share s 1

Ski

ll Sh

are

s2 d=0.8

0 10

1

Skill Share s 1

Ski

ll Sh

are

s2 d=0.6

0 10

1

Skill Share s 1

Ski

ll S

hare

s2 d=0.4

0 10

1

Skill Share s 1

Ski

ll S

hare

s2 d=0.2

Integration and Stereotypes

Definition: Integration eliminates negative stereotypesif there exists δ ∈ (0, 1) such that an asymmetric equilibrium exists iff d ≥ δ.

If there is no such δ, either discrimination persists even under complete integration or cannot arise under any level of segregation

If integration eliminates negative stereotypes it triggers shift from asymmetric to symmetric equilibrium

This can raise skill shares in both populations, or lower them in both

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Curvature of Neighborhood Effects

Under what conditions can integration eliminate negative stereotypes?

Two things matter: population composition β and curvature of neighborhood effects a

When a is small (large), increases in neighborhood quality have sharpest effect on costs when initial neighborhood quality is high (low)

Integration favors disadvantaged group most (and hurts advantaged group least) when a is small

Effects of Integration

For any population composition β, there exist values of a such that integration eliminates negative stereotypes

For this to happen, a must be sufficiency small or sufficiently large

The thresholds am(β) and an(β) are decreasing in βThere can exist values of a such that

... integration eliminates stereotypes

... if either population share is sufficiently small

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0

1

Curvature of Neighborhood Effects

Pop

ulat

ion

Sha

re o

f Dis

adva

ntag

ed G

roup

am(β)

an(β)

Welfare EffectsIf integration eliminates negative stereotypes

Then it is welfare-enhancing if ... neighborhood effects are strongest ... at low levels of human capital... and share of disadvantaged group is sufficiently small

And it is welfare-reducing if ... neighborhood effects are strongest ... at high levels of human capital... and share of disadvantaged group is sufficiently large

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ConclusionsImplications of integration span multiple marketsSegregation affects possibility and extent of

statistical discrimination Integration can eliminate negative stereotypes, but

may enhance or reduce welfare In areas in which stereotyped group is small ... if human capital spillover effects are most

powerful at low levels of skill accumulation ... integrationist policies should enjoy popular

support

Stereotypes & Crime

(with Brendan O’Flaherty)

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Robbery

Robbery is costly, widespread, and often violent In 2002 alone in the United States

... around half-a-million robberies

... at an aggregate cost to victims of $5.4 billion

... including property damage, medical expenses, lost productivity

... but excluding costs of precautions, fear, racial friction

... and almost 40% of (single-offender) robberies involved injury

Racial DisparitiesAfrican-Americans disproportionately likely to be

victims/arrestees/prisonersTwice as likely to be victims, eight times as likely

to be arrested (relative to whites, 2002) Sixteen times as likely to be incarcerated in state

prison (relative to non-Hispanic whites, 2002)Around 151,000 incarcerated, of whom 91,000

black (2002) 16.6% of African-Americans in prison have been

convicted of robbery

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Racial Disparities (continued)

1.0000.1170.4570.426Total0.2660.0130.2260.027BlackVictim0.7340.1040.3310.399WhiteTotalOtherBlackWhite

Perceived race of robber

Why the disparity?Why are white-on-black robberies so rare, and

black on white so common?

This runs counter to common beliefs about racism: ... if whites dislike blacks ... or law enforcement undervalues black safety ... or courts discount black testimony ... especially against whites ... then white criminals (who are plentiful) ... should eagerly rob blacks

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What is Robbery?In the US, robbery is defined as "taking, or attempting to

take, anything of value from the care, custody, or control of a person or persons by force or threat of force or violence and/or by putting the victim in fear“

Examples: muggings, hold-ups Unsuccessful attempts count as robberies Robbery involves use or threat of force (burglary and theft

do not) Personal confrontation is a necessary element: stealing an

empty car is motor vehicle theft; carjacking is robbery But personal confrontation is not sufficient: barroom

brawls and domestic violence are assaults, not robberies

Key IdeasRobberies involve sequence of decisions based on

incomplete information Robber decides whether to confront a given victim Victim decides whether to resist or comply Robber decides whether to flee or use force Robbers assess likelihood of resistance Victims assess likelihood of violence (conditional

on resistance)Such probability assessments are likely to be race-

contingent

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Race-Contingent Beliefs & Actions

If victims believe black offenders are more likely to force compliance ... they will be less likely to resist ... making robbery more lucrative for blacks ... including those who are non-violent

If offenders believe black victims are more likely to resist ... they will prefer white victims

How might such beliefs arise in equilibrium, without innate group differences in propensity for violence?

Inequality and IncentivesThis can happen if

... probabilities of victim resistance and offender violence

... are correlated with characteristics such as wealth

... which are unobservable to participants in a robbery

... but which exhibit systematic differences across groups

Victims and robbers engage in statistical discrimination ... Resulting in racial disparities in equilibrium behavior ... and hence in rates of resistance, violence, and crime

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Incomplete InformationEach player is drawn from a set of types Θ = [θmin,θmax] Interpretation: outside option or initial wealth θ describes characteristics unobservable to participants in

robbery

Given observable characteristics of victim and robber ... distribution of robber types given by F(θ) : Θ → [0,1] ... distribution of victim types given by G(θ) : Θ → [0,1]

F and G are continuous and commonly known Payoffs xi(θ) , yi(θ) , and zi(θ) are differentiable

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Key Assumptions

Robbers prefer successful robbery to no attempt to unsuccessful attempt

Victims prefer no robbery to unsuccessful attempt to successful robbery to violence

Wealthier victims lose less when robbedWealthier victims lose more from violenceWealthier robbers lose more from violence

Perfect Bayesian Equilibria

The common priors F(θ) and G(θ)... with the functions xi(θ), yi(θ) and zi(θ)... define an extensive-form Bayesian game

Focus on Perfect Bayesian Equilibria: Strategies are optimal conditional on beliefs Beliefs are correct conditional on strategies

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Notation

Given F(θ) and G(θ), let µ denote probability of resistance by victim (robber belief)

Let λ denote the probability of violence, given resistance (victim belief)

Equilibrium beliefs (λ∗,µ∗) must be such that ... the optimal response to these beliefs results in... probabilities of resistance and violence ... that are consistent with these beliefs

Crime RatesGiven payoff and distribution functions, there is a unique

set of equilibrium beliefsHence rates of crime, violence and resistance are uniquely

determined Three types of crime arise in equilibrium: nonviolent

robberies, violent robberies, failed robbery attemptsOverall crime rate (aggregating three crimes) is γ∗Proportion of attempts successful without violence: 1-µ∗

Proportion of attempts that fail: µ∗(1-λ∗) Proportion of attempts that end in violence: µ∗λ∗

These rates can be recovered from data

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Racial DisparitiesSuppose each individual belongs to one of two groupsGroups differ with respect to their type distributions Let Fb(θ) and Fw(θ) denote robber distribution functions Gb(θ) and Gw(θ) denote victim distribution functionsPayoff functions xi(θ), yi(θ), zi(θ) identical across groups

For any given interaction ... once race of victim and offender have been observed... equilibrium behavior is uniquely determined

This behavior will generally be race-contingent

NotationLet λij denote victim's belief that resistance will be met

with violence when offender belongs to group i and victim to group j

Let µij denote robber's belief that victim will resist when offender belongs to group i and victim to group j

For each i,j pair, equilibrium beliefs (λij∗,µij

∗) must satisfy: ... optimal response to beliefs by victims and robbers ... results in objective probabilities of resistance/violence ... that are consistent with these beliefs

Resulting crime rate (offender in group i and victim in group j) is γij∗

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Main Results (fixed victim)

Holding constant the race of victim

Black crime rates higher than white crime rates

Black offenders face less resistance

Black offenders respond violently to resistance at higher rate

Main Results (fixed offender)

Holding constant the race of the offender

White victims confronted at higher rates than black victims

White victims resist at lower rates

Black victims face higher likelihood of violence conditional on resistance

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Implications for Crime Rates

Taken together, these results imply:γwb < min{γww, γbb} ≤ max{γww, γbb} < γbw

Do the data support this prediction? For 2002, assuming random matching ... which overpredicts within-group crime rates... and normalizing γww = 1 we get:

γwb = 0.21 < 4.74 = γbbwhich is statistically significant

Violence

Prediction: holding constant offender race, white victims face lower probability that resistance will be met with violence

Prediction (for black offenders): λbw < λbb

Aggregating over 1993-2002, we obtain:λbw = 0.430 < 0.608 = λbb

which is statistically significant

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Resistance

Prediction: all victims resist with greater likelihood when faced with white (rather than black) offender

Prediction (with white victims): µww > µbw

Aggregate estimates of resistance 1993-2002:µww = 0.612 > 0.550 = µbw

which is statistically significant

Disaggregated Data

0.6730.6740.5630.4990.4660.7030.6660.4910.6990.724µww

0.4110.5490.4420.6250.5120.5920.5230.5700.6210.561µbw

2002200120001999199819971996199519941993

0.6990.5220.6980.6590.5180.7270.5350.6520.6460.461λbb

0.7760.7540.6940.3240.4800.5630.2960.2420.3730.322λbw

2002200120001999199819971996199519941993

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ConclusionsGroup inequality affects incentives, inducing otherwise

identical individuals to behave differently in equilibriumVictims believe black offenders more likely to respond

violently to resistance, which lowers incentives to resist and makes crime more lucrative for black offenders

Robbers believe black victims more likely to resist robbery attempts, which makes them less attractive targets

Implication: black victims face a group of offenders that is on the whole more violent... this prediction is roughly consistent with empirical record

Extensions

Racial stereotypes arising from group inequality... affect individual incentives... inducing otherwise identical individuals... to adopt different equilibrium behavior

This phenomenon is most relevant... when relevant information is limited... and actions have major consequences

Could this account for residential segregation?