The Effect of Underclass Social Isolation on Schooling Choicesites.duke.edu/econ206_01_s2011/files/2012/05/13...The paper also shows that the introduction of more representa-tive role

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PETER STREUFERTUniversity of Western Ontario

Abstract

This paper models underclass social isolation as the loss of high-income role models and then studies the plausible conjecture thatthis isolation depresses the level of schooling chosen by under-class youth. It is found that although this conjecture stands inroughly calibrated simulations, it is not a theoretical necessity.The paper also shows that the introduction of more representa-tive role models during university can polarize underclass youthand that income support programs depress schooling choicethrough a distinct and reinforcing channel.

1. Introduction

This paper studies a central thesis of Wilson’s ~1987! book The TrulyDisadvantaged, namely, the thesis that the social isolation of the underclasscauses underclass youth to underestimate the effect of schooling on incomeand consequently, to choose too little schooling.

When he states this thesis most precisely ~pp. 56–58!, Wilson arguesthat “a perceptive ghetto youngster in a neighborhood that includes agood number of working and professional families . . . can see a connec-tion between education and meaningful employment.” Yet, “in a neigh-borhood with a paucity of regularly employed families . . . the relationshipbetween schooling and postschool employment takes on a different mean-ing.” This results in “a shockingly high degree of educational retardation.”

Department of Economics, University of Western Ontario, London, Ontario, N6A 5C2 Can-ada [email protected]!. I have received valuable comments from the editors and refer-ees of this journal and from seminar participants at Wisconsin, Vanderbilt, Purdue, Boston,Albany, Texas, Stanford, and Northwestern. I also thank the Institute for Research on Pov-erty at the University of Wisconsin for its financial support, Wayne Sigelko for his insightfulresearch assistance, and my father, Victor Streufert, for the example set by his own work onthe sociology of urban poverty ~Streufert, 1972!.

Received 30 September 1998; Accepted 5 January 20000.

JPET 24-3

© 2000 Blackwell Publishers, Inc.Journal of Public Economic Theory, 2 ~4!, 2000, pp. 461–482.

461

We begin by modeling how role models shape a young person’s per-ception of the relationship between schooling and income. A role modelis taken to be a single observation of the schooling and income of anadult worker, and many such observations enable a young person to ~non-parametrically! regress income on schooling. Thus a young person’sinformation-gathering process resembles a labor economist’s estimationof the earnings function. If the role models that a young person observesare representative of the labor force, she will be well informed whenmaking her schooling decision. Section 2 defines schooling choice in thisideal “benchmark” model, provides a useful incremental characterizationof this choice ~Theorem 1!, and roughly calibrates a “Benchmark Simula-tion” to fit Mincer’s earnings function and the schooling distribution ofthe U.S. labor force.

Underclass social isolation is then modeled as the elimination of high-income observations at each level of schooling. That is, we assume that anunderclass youth observes a sample of role models that is truncated fromabove. Wilson’s thesis suggests that the resulting bias in the regressiondepresses schooling.

Section 3’s “Isolation Simulation” accords with this thesis. In it, ayoung person who completed high school in the Benchmark Simulationonly completes grade eight. This precipitous drop in her schooling resultsfrom the lack of any role models earning more than $15,0000year. Aneconomist would say that the representative role models she lacks definea public good that is lacking in an underclass neighborhood. Alternatively,a sociologist might be pleased to hear an economist argue that a “ratio-nal” individual’s schooling decision depends critically on that individual’ssocial environment.

However, Section 3’s “Theoretical Counterexample” shows that socialisolation need not depress schooling. If Wilson’s conjecture stands, itstands because social isolation reduces the perceived increment to incomethat would result from further schooling. But this need not be the case, asdemonstrated by the counterexample. Furthermore, social isolation inev-itably causes a young person to underestimate the income she forgoeswhile attending school, and this factor can only serve to make schoolmore attractive. Thus, social isolation depresses schooling only if ~1! itleads to a decrease rather than an increase in perceived incrementalincome and ~2! the magnitude of this decrease is sufficiently great toovercome the underestimation of forgone income. ~In statistical terminol-ogy, truncation depresses schooling only if ~1! it decreases rather thanincreases the slope of the regression and ~2! the magnitude of this decreaseoutweighs the drop in the level of the regression.!

Section 4 extends the model by allowing a person’s collection of rolemodels to change as she proceeds through school. For example, the“Polarization Simulation” assumes that a person’s social isolation is liftedif and when she enters university. Such changing role models split under-class youth into two disparate groups: one whose schooling is stunted by

462 Journal of Public Economic Theory

underclass isolation and another that is ultimately unaffected by its under-class origins. No one in the first group starts university, while everyone inthe second group goes on to graduate school.

Section 5 alters the model to study the effects of income supportprograms. In contrast to social isolation, income support unambiguouslyreduces incremental income ~i.e., the slope of the regression! and increasesforgone income ~i.e., the level of the regression!. Thus, income supportinevitably depresses schooling ~Theorem 2!. The “Support Simulation”shows that this effect can be quite pronounced in the lower tail of thepopulation.

The “Combination Simulation” of Section 6 shows that social isolationand income support are likely to depress schooling through distinct andreinforcing channels. One squeezes the regression from above, and theother squeezes from below.

Section 7 concludes.

2. Benchmark Model

2.1 Specification

Imagine that a young person must choose a level of schooling s from the13-element set $7,8, . . . ,19%. For example, the choice s 5 18 describessomeone who chooses to enter the labor force after completing 18 grades~i.e., high school, four years of university, and two years of postgraduatetraining!, at about 24 years of age. The choice s 5 7 is less straightforwardbecause few children literally enter the labor force after completing sevengrades, at about 13 years of age. Rather, the choice s 5 7 is better inter-preted as someone who becomes unmotivated during junior high school,fails two grades, and then quits altogether as soon as she is able to do so~in the United States, this becomes legal on her 16th birthday!.

One key factor in a young person’s decision is the effect that s willhave upon her future annual income y ~for simplicity, assume y is constantthroughout her career and ignore nonwage income!. The actual relation-ship between s and y is specified by the set of cumulative distributionfunctions1 ^Fs&s[$7,8, . . . ,19% . For every schooling s and income y , Fs~ y ! is theprobability that a person with schooling s will earn an income of less thanor equal to y .

Imagine that a young person learns about ^Fs&s by observing rolemodels from the labor force. Formally, a role model is a single observa-tion of schooling s and income y . As elucidated by Manski ~1993!, thismathematical concept of role models accords with the sociological con-cept of role models, and a collection ~i.e., sample! of such role modelsaccords with the sociological concept of a reference group.

1Each Fs : R1 r @0,1# is assumed to have a finite mean.

Underclass Social Isolation and Schooling Choice 463

For simplicity, we assume asymptotic sampling. That is, we assume thatfor every s , a young person learns the entire cumulative distribution func-tion rather than needing to draw statistical inferences from a finite sam-ple. Thus, at every s , the sample mean income and the population meanincome coincide at E @Fs# . In other words, Ê @Fs#&s can be regarded as aperfect nonparametric regression of income y on schooling s .

Let d [ ~0,1! be a person’s discount factor and let u [ R specify theannual disutility a person experiences as the result of attending school.She then chooses her schooling s so as to maximize the difference betweenthe income of her work years and the disutility of her school years. For-mally, a person with disutility parameter u uses the regression Ê @Fs#&s toselect the schooling

S u ~Ê @Fs #&s ! 5 maxHarg maxH (a5s11

59

d a21E @Fs # 2 (a51

s

d a21u*s [ $7,8, . . . 19%JJ .

Note that s rather than s is used as the choice variable so as to avoidconfusion with the s in Ê @Fs#&s , that a 5 59 coincides with retirement at65 years from birth, and that the outer max operator simply means that ifseveral schoolings maximize utility then the tie is resolved in favor of thehighest schooling. This choice problem closely resembles Rosen ~1977,pp. 9–13!, and numerous variations appear elsewhere in the labor eco-nomics literature ~e.g., Heckman 1976; and Ryder, Stafford, and Stefan1976, to name but two!. Notice that schooling unambiguously declineswith the disutility parameter u but that the effects of the discount factor dand the regression Ê @Fs#&s are ambiguous.

2.2 Incremental Characterization of Schooling

Given the regression Ê @Fs#&s , we can derive the incremental benefit ofattending grade s [ $8,9, . . . ,19% as

Bs~Ê @Fs #&s ! 5 d~E @Fs # 2 E @Fs21 # !~1 2 d592s !0~1 2 d! 2 E @Fs21 # .

The first term is the present discounted value of the increase in futureincome that a person will enjoy as a result of investing in grade s asopposed to grade s 2 1. The second term is the income a person willforgo while attending grade s. The following theorem states that a personwill continue to attend school until the benefit of an additional year ofschooling falls below her disutility.

THEOREM 1: Suppose that Bs~Ê @Fs#&s! is weakly decreasing in s. Then

S u ~Ê @Fs #&s ! 5 max$7% ø $s ≥ 8 6Bs~Ê @Fs #&s ! ≥ u%.

Proof: See the Appendix.

The incremental net benefit Bs~Ê @Fs#&s! is weakly decreasing in swhenever two conditions are met: ~1! that the increment to future income


is declining with each additional year of schooling, in other words, thatÊ @Fs#&s is concave:

~ ∀s [ $9,10, . . . 19%!E @Fs # 2 E @Fs21 # ≤ E @Fs21 # 2 E @Fs22 #

and ~2! that forgone income is increasing with each additional year ofschooling, in other words, that Ê @Fs#&s is increasing.

However, Bs~Ê @Fs#&s! can be weakly decreasing in s even when theincrement to future income is not decreasing ~as, e.g., in the BenchmarkSimulation!. This occurs when the increase in the ~discounted! incrementto future income is outweighed by the increase in forgone income ~whichis more likely to happen when the discount factor is low!. This discussionis important because Ê @Fs#&s is convex in the large empirical literaturethat linearly regresses the logarithm of income on schooling.

2.3 Benchmark Simulation

Parameters. Each person’s optimization problem has three parameters: thenonparametric regression Ê @Fs#&s , the discount factor d, and the disutilityparameter u. All people share the same ^Fs&s and d, while each has herown unique u.

For every y , specify Fs as the lognormal cumulative distribution func-tion with mean E @Fs# satisfying

(a5s11

59

d a2~s11!E @Fs # 5 (a5s11

59

d a2~s11!y ~s ,a 2 s !,

where

y ~s , x ! 5 4480~4.87 1 0.255s 2 0.0029s2 2 0.0043sx 1 0.148x 2 0.0018x 2 !,

and with standard deviation equal to half of this mean. The first equationstates that E @Fs# is the present discounted value of lifetime earnings,expressed at an annual rate. The second equation calculates each year’searnings as a function of schooling s and experience x 5 a 2 s . Thissecond equation is the earnings function estimated by Mincer ~1974!,discussed by Willis ~1986, Table 10.5!, and crudely adjusted for inflation~so that income is measured in $0year!. Note that Ê @Fs#&s is depicted inFigure 1a as the solid curve.

Specify the discount factor as d 5 0.85. This discount rate can beregarded as modeling not only time preference but also financial con-straints. A discount factor of 0.85 corresponds to a discount rate of 17%.As the rate of return to schooling is about 17% at s 5 8 ~Willis 1986,p. 546!, significantly lower discount rates result in no schooling choicesnear s 5 8.

The simulation concerns 100 individuals that differ only in their dis-utility parameter u. The 100 values chosen can be seen as the vertical


coordinates of the 100 stars in Figure 1c. These 100 us are “log-beta-ly”distributed: the base-2 logarithms of a linear transformation of the 100 u’sare percentiles from a symmetric beta distribution ~the remarks in theprocedure maketv of Streufert 1999 give full details!. We chose this distri-bution of us so that the resulting distribution of schooling would roughly

a b

c

Figure 1: Benchmark simulation. Figure 1a’s curve depicts the true relationshipbetween schooling and expected income. On the basis of this, the 100 personsdistinguished by their disutility parameters u make the schooling choices shown byFigure 1c’s stars. Figure 1a’s stars depict the random incomes they receive as aconsequence, and Figure 1b shows the aggregate distribution of income that results.


match the actual distribution of schooling ~U.S. Bureau of the Census1990, Table 632!. Most individuals have been assigned negative disutilities:this should be interpreted to mean that most individuals find school lessonerous than work.

Results. Each of the 100 individuals is depicted by a star in Figure 1c. Thestar’s vertical coordinate gives her disutility parameter u, and its horizon-tal coordinate shows her chosen schooling S u~Ê @Fs#&s!. For example,persons u 5 $4020year and u 5 $3010year have been assigned the greatestdisutility parameters and only complete grade seven. Meanwhile person u5 $2290year has been assigned the third-highest disutility parameter andcompletes grade eight. In accord with Theorem 1, their decisions arecompletely explained by the fact that the incremental net benefit to attend-ing grade eight, namely, B8~Ê @Fs#&s!, lies below $301 and above $229. Ingeneral, the incremental net benefit Bs~Ê @Fs#&s! lies just slightly abovethe top star at schooling s. Note that Bs~Ê @Fs#&s! is decreasing in s, asassumed by Theorem 1.

As a consequence of their chosen schooling s , each person receives anincome y drawn from the distribution Fs . This is depicted by the 100 starsin Figure 1a. For example, persons u 5 2$9,9510year and u 5 2$11,8010year ~the two persons with the lowest disutility parameters! chose 19 yearsof schooling, and their two incomes are depicted by the two stars at s 5 19in Figure 1a. In particular, these stars are at the 25th and 75th percentilesof the distribution F19 ~they are located asymmetrically around the meanE @F19# because F19 is a lognormal distribution!.

Figure 1b shows the density of the aggregate distribution of incomethat results from the schooling decisions shown in Figure 1c. Crudelyspeaking, this density is found by sweeping all of the stars in Figure 1a tothe vertical axis and letting them pile up.

3. Social Isolation

3.1 Theory

We model underclass social isolation by assuming that an underclass youthobserves, for each schooling s , a distribution of incomes that is truncatedfrom above by a. This truncation models selective out-migration from anunderclass neighborhood: everyone with an income above a leaves theghetto, while everyone with an income at or below a remains in theghetto and provides a role model for underclass youth.

Formally, let E @Fsa# denote the expectation of the distribution Fs

a

that is obtained by truncating2 the distribution Fs from above at a ≥ 0. Aperson with disutility parameter u observing the truncated regressionÊ @Fs

a#&s will choose schooling S u ~Ê @Fsa#&s !. Wilson’s conjecture is that

2That is, define Fsa ~ y ! 5 1 if y . a and Fs

a ~ y ! 5 Fs~ y !0Fs~a! if y ≤ a.


truncation discourages schooling, in other words, that a decrease in awill cause a decrease in S u ~Ê @Fs

a#&s !. This does occur in the IsolationSimulation of Section 3.2. However, this is not a logical necessity, asdemonstrated by the Theoretical Counterexample of Section 3.3.

3.2 Isolation Simulation

Parameters. The Isolation Simulation alters the Benchmark Simulation bytruncating every distribution Fs from above at a 5 $15,0000year. Thismeans that no young person ever observes anyone earning more than$15,0000year. ~This cutoff happens to be above both the mean and themedian of true income distribution F12 .!

Results. Figure 2a depicts the $15,0000year truncation ~solid line! whichshifts the regression down from Ê @Fs#&s to Ê @Fs

15000 #&s ~i.e., from thedotted curve to the solid curve!.

Figure 2c shows that this misperception shifts schooling back ~fromthe small stars to the large stars!. For example, consider the person withthe third highest disutility parameter, that is, person u 5 $2290year. In theBenchmark Simulation ~small stars in Figure 2c!, she chose eight grades.Here in the Isolation Simulation ~large stars in Figure 2c!, she choosesseven grades. She does this because the perceived incremental benefit ofgrade eight has fallen from something above $229 to something ~far!below $229.

The perceived incremental benefit of grade eight can be determinedmore precisely. In the Benchmark Simulation, it was between $301 and$229 because person u 5 $3010year chose grade seven and person u 5$2290year chose grade eight. Here, it lies between 2$1,1200year and2$1,1650year because person u 5 2$1,1200year chooses grade seven andperson u 5 2$1,1650year chooses grade eight.

Something similar is occurring at all grade levels. The incrementalbenefit of grade s is approximately equal to the highest star at grade s.Isolation has caused this incremental benefit to shift down. By Theorem 1,this is equivalent to shifting back schooling. All this can be seen in Fig-ure 2c: the downward shift of the highest stars explains the backward shiftof every person’s schooling choice.

The consequences of this backward shift in schooling are drawn outin the remaining diagrams. Figure 2d shows the ~lifetime! utility loss thateach person suffers, and Figure 2b shows the downward shift in the incomedistribution. The stars in Figure 2a depict the incomes that the 100 per-sons receive. Twelve stars lie above the truncation line. This suggests that12 persons will escape the ghetto, while the remaining 88 will stay behindand provide a truncated sample of role models for the next generation~this sentence is not formally consistent with the model since the incomecoordinates of Figure 2a’s stars are percentiles rather than random vari-ables and since the next generation will draw an asymptotic rather thanfinite sample!.


3.3 Theoretical Counterexample

As suggested by the Isolation Simulation ~and as formally proven in Theo-rem 1 under the assumption that the incremental benefit of grade s

a b

c d

Figure 2: Isolation simulation. Figure 2a depicts the $15,000 truncation ~solid line!

that shifts the regression down ~from dotted curve to solid curve!. Figure 2c showsthat this shifts the schooling of the 100 persons back ~from small stars to big stars!.Consequently, they earn the incomes depicted in Figure 2a ~big stars!, they loselifetime utility as shown in Figure 2d ~big stars relative to zero loss at little stars!,and Figure 2b’s aggregate distribution shifts down ~from dotted curve to solidcurve!.


decreases with s!, truncation will depress schooling precisely when itdepresses the perceived incremental benefit

Bs~Ê @Fsa#&s ! 5 d~E @Fs

a# 2 E @Fs21a # !~1 2 d592s !0~1 2 d! 2 E @Fs21

a # .

Truncation need not depress the incremental benefit: the effect of trun-cation on the first term is ambiguous, and its effect on the second term isunambiguously in the opposite direction.

Begin with the second term. E @Fs21a # is the level of the regression at

s 2 1, and it is the income one forgoes while attending grade s ratherthan starting work after grade s 2 1. Truncation must depress one’sperception of forgone income, and this can only serve to increase theperceived incremental benefit of grade s.

The first term concerns the slope of the regression line between s 5s 2 1 and s 5 s, namely, E @Fs

a# 2 E @Fs21a # . This slope is the incremental

income one receives in each working year as a result of attending grade srather than grade s 2 1. Although it is plausible that truncation willf latten the regression line as it did in the Isolation Simulation, this is nota theoretical necessity.

This conceptual possibility is illustrated by the Theoretical Counter-example. In this simulation, each Fs is defined to place one-third of theprobability at each of the three triangles shown at s in Figure 3a. At everys , one triangle is located at $50,0000year, a second is located at $30,0000year, and the third is located at s times $1,0000year. Truncation at $20,0000year ~Figure 3a’s f lat solid line! lops off the top two triangles from everyFs , and this serves to accentuate the differences between the distributions.As a result, the truncated regression ~solid upward-sloping line! is steeperthan the true regression ~dotted upward-sloping line!.

Thus, in the Theoretical Counterexample, truncation leads one tounderestimate the income forgone by another year of schooling and tooverestimate the incremental income that will result. Both factors leadone to overestimate the incremental benefit of schooling.

Figure 3b’s small stars show that all persons would choose seven gradeswhen perceiving the true regression. This implies that the true incremen-tal benefit of grade eight is less than the lowest disutility parameter of u 52$11,8010year. Figure 3b’s large stars show that everyone with disutilityparameter less than u 5 2$1,3000year would choose more schooling if theymisperceived the regression because of truncation. This implies that theincremental benefit of grade eight has risen to at least 2$1,3000year becauseof truncation.

4. Polarization

4.1 Theory

This section modifies the model of the previous section by letting thetruncation cutoff a vary as an underclass youth moves through school.


Her schooling decision then becomes a sequential problem: she continueson in school if and only if the role models that she has observed up untilthat time suggest that some further schooling is better than stoppingimmediately.

It is reasonable to assume that a increases over time, for this wouldmodel the notion that, as an underclass youth moves from grade school to

a

b

Figure 3: Theoretical counterexample. Figure 3a depicts a $20,000/year trunca-tion ~solid flat line! that shifts the regression down ~from dotted upward-sloping lineto solid upward-sloping line! while increasing its slope. Figure 3b shows that thisincreases the schooling of 59 persons ~from small stars to big stars! because thetruncation causes them to overestimate the benefit of schooling.


high school to university and finally to graduate school, the social isola-tion of her underclass origins is gradually lifted. Unfortunately, early socialisolation might so depress her schooling choice that she drops out beforeobserving the new role models.

Formally, let ât &t[$7,8, . . . ,19% be a sequence of cutoffs. We assume thata person with disutility parameter u and truncation cutoffs ât &t[$7,8, . . . ,19%

will choose schooling

min$t ≥ 7 6Stu ~Ê @Fs

at #&s ! 5 t %,

where

Stu ~Ê @Fs

at #&s !

5 max Harg max H (a5s11

59

d a21E @Fsat # 2 (

a51

s

d a21 u*s [ $t , t 1 1, . . . ,19%JJ .

The first expression says that a person will stop attending school in the firstgrade t such that the role models she is viewing under the truncation cutoffat lead her to choose t grades of schooling. The second expression definesthe symbol St in a manner that is identical to S except for the constraint thata person cannot choose a level of schooling lower than t ~thus if she doesn’tchoose t grades she is choosing to continue rather than regress!.

Note that a person does not anticipate that her role models mightchange and therefore, she has no incentive to continue in school for thesake of gathering information ~Manski and Wise 1983, p. 10!.

4.2 Polarization Simulation

Parameters. This simulation differs from the Isolation Simulation in that aperson’s isolation ends abruptly when she enters university. Formally, at 5$15,0000year for t ≤ 12, and at 5 1$`0year for t . 13.

Results. See Figure 4. This simulation is essentially an amalgamation of theIsolation and Benchmark Simulations: isolation causes an underclass youthto choose too little schooling ~as in the Isolation Simulation! unless shechooses to begin university in spite of her isolation, in which case shechooses an efficient level of schooling ~as in the Benchmark Simulation!on the basis of role models observed after entering university.

This discontinuous change in information divides underclass youthinto two classes: school-loving youth ~with disutility parameters u , 2$7,0000year and postgraduate training! who are ultimately unaffected by theisolation of their ghetto origins and their comparatively school-aversepeers ~with disutility parameters u . 2$7,0000year and no more than ahigh-school education! who never overcome their isolation. This tendencyto polarization accords with empirical findings that the income distribu-tion of black men has become polarized ~Murray 1984, p. 92 ~citing Kilson1981!; and Wilson 1987, p. 45 ~citing Levy 1986!!.


5. Income Support

5.1 Theory

Return to the benchmark model. Then modify it by introducing a stylizedincome support program in which everyone who has left school is guar-

a b

c d

Figure 4: Polarization simulation. Given that a person’s isolation ends upon enter-ing university, persons with the lowest disutility parameters ~u , −$7,000/year!are ultimately unaffected by their isolated origins and attend graduate school,while the remainder are harmed as they were in the Isolation Simulation and neverenter university. ~As before, small stars and dotted lines recall the BenchmarkSimulation.!


anteed an income of at least b. This implies that an underclass youth willobserve no role models with an income below b and will observe anappreciable number of role models earning exactly b.

Formally, let E @Fs , b# denote the mean of the distribution Fs , b derivedfrom Fs by censoring3 from below at b ≥ 0. Then suppose that a personwith disutility parameter u uses the regression Ê @Fs , b#&s to choose theschooling S u~Ê @Fs , b#&s!, where S is defined as in the benchmark model.

The following theorem shows that support must depress schooling.The gist of its proof concerns the effect of support on the incrementalbenefit of grade s, namely,

Bs~Ê @Fs , b #&s ! 5 d~E @Fs, b # 2 E @Fs21, b # !~1 2 d592s !0~1 2 d! 2 E @Fs21, b # .

Support must diminish incremental future income ~the first term, which isthe slope of the regression! and must increase forgone income ~the mag-nitude of the second term, which is the level of the regression!. Botheffects decrease the incremental benefit of grade s and thereby depressschooling.

THEOREM 2: Suppose that ~ ∀s [ $8,9, . . . ,19%!Fs first-order stochastically dom-inates Fs21. Then b ' ≥ b implies S u ~Ê @Fs , b ' #&s ! ≤ S u~Ê @Fs , b#&s!.


Furthermore, an income support program harms society as a whole inthe sense of the following theorem’s inequality. Its left-hand side is theutility gained by a person as a consequence of income support b. Itsright-hand side is the government’s cost of the support provided to thisperson. The two are comparable because a person’s utility is measured inunits of income.

THEOREM 3: Let s * 5 S u~Ê @Fs#&s! be schooling in the benchmark model andlet s 5 S u~Ê @Fs , b#&s! be schooling given income support b. Then

S (a5s11

59

d a21E @Fs, b # 2 (a51

s

d a21 uD2S (a5s *11

59

d a21E @Fs * # 2 (a51

s *

d a21 uD≤ (

a5s11

59

d a21 ~E @Fs, b # 2 E @Fs #!.


5.2 Support Simulation

Parameters. The Support Simulation alters the Benchmark Simulation bycensoring every distribution Fs from below at b 5 $5,0000year. This means

3That is, define Fs , b~ y ! 5 0 if y , b and Fs , b~ y ! 5 Fs~ y ! if y ≥ b.


that every role model who would have earned less than $5,0000year isinstead making $5,0000year as a consequence of the income support pro-gram. ~This guaranteed income happens to be below both the mean andthe median of the true income distribution F7 .!

a b

c

Figure 5: Support simulation. Figure 5a depicts a $5,000/year income support~solid line! that shifts the regression up ~from dotted curve to solid curve!. Figure 5cshows that this causes 34 persons to complete only grade seven. Consequently,they earn the ~unsupported! incomes depicted at s 5 7 in Figure 5a and causeFigure 5b’s aggregate income distribution to shift down ~from dotted curve to solidcurve!.


Results. Theorem 2 states that support must depress schooling. This sim-ulation shows that the effect can be rather severe under a modest supportprogram that ensures that everyone makes at least $5,0000year. Figure 5c’slarge stars show that 34% of the simulation’s hundred persons completeonly seven grades. In marked contrast, Figure 5c’s small stars show thatonly two persons made this choice in the Benchmark Simulation.

While the Isolation Simulation showed that social isolation depressesalmost everyone’s schooling choice, this simulation suggests that supportaffects mainly the lower portion of the population. The most severelyaffected person has disutility parameter u 5 2$9960year. She completesonly seven grades even though she would have chosen 11 grades in theabsence of income support ~Figure 5c!. ~There is no figure depictingutility loss. Support benefits every individual ~if they don’t have to pay forit! and harms society as a whole ~in the sense of Theorem 3!.!

6. Combination Simulation

Parameters. This simulation combines the Polarization and Support Simu-lations. In particular, it models social isolation with at 5 $15,0000year fort ≤ 12 and at 5 1$`0year for t ≥ 13, and it models an income supportprogram with b 5 $5,0000year.

Results. See Figure 6. Because social isolation and income support squeezethe regression line from opposite sides, one might anticipate that the twodepress schooling through distinct and reinforcing channels. This simu-lation accords with that intuition. No person chooses more schooling herethan they did in either the Polarization Simulation or the Support Simu-lation, and 43 persons choose less than they did in either. For example,person u 5 2$2,7520year chose 14 grades in the Benchmark Simulation,nine grades in the Isolation and Polarization Simulations, 14 grades in theSupport Simulation, and seven grades here in the Combination Simulation.

7. Conclusions

Although the effect of social isolation on schooling choice is ambiguousfrom a theoretical perspective, our simulations suggest that it can mark-edly depress schooling and that it can divide underclass youth into twopolarized classes. Social isolation would be reduced if geographical neigh-borhoods were made more representative of the entire society. But, it maybe more efficacious to focus on other sorts of “neighborhoods,” such asnetworks of family or friends, or organizations concerned with cultural orrecreational pursuits. It seems that disembodied information about thetrue relationship between schooling and income is not easily transmitted.

Income support inevitably depresses schooling, and our simulationssuggest that it affects primarily the lower portion of the population. This


is not a matter of misinformation. It is an unintended consequence ofincome support.

Finally, it is hoped that this paper and its computer programs willinform empirical studies of the relationship between social isolation andschooling choice. The fundamental factor is more complex than meanneighborhood income. It is the representativeness of the adults with whichyoung people can identify.

a b

c

Figure 6: Combination simulation. Social isolation and income support depressschooling through distinct and reinforcing channels. When childhood role modelsearn less than $15,000/year and a support program guarantees an income of$5,000/year, 66 of the 100 persons complete only seven grades ~big stars inFigure 6c!.


Appendix

Proof of Theorem 1: Since the following statements are equivalent by alge-braic manipulation, we conclude that s is weakly preferred to s 2 1if and only if Bs~Ê @Fs#&s! ≥ u.

(a5s11

59

d a21E @Fs # 2 (a51

s

d a21 u ≥ (a5s

59

d a21E @Fs21 # 2 (a51

s21

d a21 u,

(a5s11

59

d a21E @Fs # 2 ds21 u ≥ (a5s11

59

d a21E @Fs21 # 1 ds21E @Fs21 # ,

(a5s11

59

d a21 ~E @Fs # 2 E @Fs21 # ! 2 ds21E @Fs21 # ≥ ds21 u,

(a5s11

59

d~a21!~s21! ~E @Fs # 2 E @Fs21 # ! 2 E @Fs21 # ≥ u,

(q51

592s

d q ~E @Fs # 2 E @Fs21 # ! 2 E @Fs21 # ≥ u,

d~E @Fs # 2 E @Fs21 # !~1 2 d592s !0~1 2 d! 2 E @Fs21 # ≥ u,

Bs~Ê @Fs #&s ! ≥ u.

For notational ease, define s * 5 S u~Ê @Fs#&s!. Note that

s * [ $7% ø $s ≥ 8 6Bs~Ê @Fs #&s ! ≥ u%, ~1!

for if s * ≥ 8, its optimality implies that it must be weakly preferred tos * 2 1 and hence that Bs * ~Ê @Fs #&s ! ≥ u by the theorem’s first sen-tence. On the other hand, since s * is defined to be the highestschooling that maximizes utility, it must be that s * 1 1 is not weaklypreferred to s * and hence that Bs *11~Ê @Fs #&s ! , u by the proof’s firstsentence. Since Bs~Ê @Fs#&s! is assumed to be weakly decreasing in s,this implies that

~ ∀s . s * !Bs~Ê @Fs #&s ! , u. ~2!

Equations ~1! and ~2! together imply that s * 5 max$7% ø $s ≥8 6Bs~Ê @Fs#&s! ≥ u%. n

Proof of Theorem 2: This paragraph shows that

~ ∀b ' ≥ b!~ ∀s1 . s!,

E @Fs1, b ' # 2 E @Fs, b ' # ≤ E @Fs1, b # 2 E @Fs, b # ,


~intuitively, this means that increased support diminishes incrementalincome!. The inequality is derived from

~ ∀b ' ≥ b!~ ∀s1 . s!

E @Fs1, b ' # 2 E @Fs1, b #

5 Hb ' 1Eb '

1`

~1 2 Fs1~ y !!dyJ 2 Hb 1Eb

1`

~1 2 Fs1~ y !!dyJ5 b ' 2 b 2E

b

b '

~1 2 Fs1~ y !!dy

5Eb

b '

Fs1~ y !dy

≤ Eb

b '

Fs~ y !dy

5 b ' 2 b 2Eb

b '

~1 2 Fs~ y !!dy

5 Hb ' 1Eb '

1`

~1 2 Fs~ y !!dyJ 2 Hb 1Eb

1`

~1 2 Fs~ y !!dyJ5 E @Fs, b ' # 2 E @Fs, b # ,

where the inequality follows from the theorem’s assumption that Fs1

first-order stochastically dominates Fs .Now define s 5 S u~Ê @Fs , b#&s!. Then

~ ∀b ' ≥ b!~ ∀s1 . s!

H (a5s111

59

d a21E @Fs1, b ' # 2 (a51

s1

d a21uJ 2 H (a5s11

59

d a21E @Fs, b ' # 2 (a51

s

d a21uJ5 (

a5s111

59

d a21 $E @Fs1, b ' # 2 E @Fs, b ' #% 2 (a51

s1

d a21 u

2 (a5s11

s1

d a21E @Fs, b ' # 1 (a51

s

d a21 u


≤ (a5s111

59

d a21 $E @Fs1, b # 2 E @Fs, b #% 2 (a51

s1

d a21 u

2 (a5s11

s1

d a21E @Fs, b ' # 1 (a51

s

d a21 u

≤ (a5s111

59

d a21 $E @Fs1, b # 2 E @Fs, b #% 2 (a51

s1

d a21 u

2 (a5s11

s1

d a21E @Fs, b # 1 (a51

s

d a21 u

5 H (a5s111

59

d a21E @Fs1, b # 2 (a51

s1

d a21 uJ2 H (

a5s11

59

d a21E @Fs, b # 2 (a51

s

d a21 uJ, 0,

where the first inequality follows from the preceding paragraph ~i.e.,support diminishes incremental income!, the second inequality fol-lows from E @Fs, b ' # ≥ E @Fs, b# ~i.e., support increases forgone income!,and the third inequality follows from the fact that s 5 S u~Ê @Fs , b#&s!~because the penultimate line is the objective at s1 given b less theobjective at s given b!. This last inequality is strict because the defi-nition of S breaks ties in favor of the highest of the maximizingschooling levels. Since the first line in the entire inequality is theobjective at s1 given b ' less the objective at s given b ', it follows thatno s1 . s can equal S u ~Ê @Fs , b ' #&s !. Hence S u ~Ê @Fs , b ' #&s ! ≤ s 5S u~Ê @Fs , b#&s!. n

Proof of Theorem 3: The theorem’s left-hand side less its right-hand side isnonpositive:

S (a5s11

59

d a21E @Fs, b # 2 (a51

s

d a21 uD2 S (

a5s *11

59

d a21E @Fs * # 2 (a51

s *

d a21 uD2 (

a5s11

59

d a21 ~E @Fs, b # 2 E @Fs #!


5 S (a5s11

59

d a21E @Fs, b # 2 (a51

s

d a21 uD2 S (

a5s *11

59

d a21E @Fs * # 2 (a51

s *

d a21 uD2 S (

a5s11

59

d a21E @Fs, b # 2 (a51

s

d a21 uD1 S (

a5s11

59

d a21E @Fs # 2 (a51

s

d a21 uD5 S (

a5s11

59

d a21E @Fs # 2 (a51

s

d a21 uD2 S (

a5s *11

59

d a21E @Fs * # 2 (a51

s *

d a21 uD≤ 0.

The two equalities are derived by algebra. The final inequality followsfrom s * 5 S u~Ê @Fs#&s! because the penultimate line is the objectiveat s less the objective at s *. n

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The Effect of Underclass Social Isolation on Schooling Choicesites.duke.edu/econ206_01_s2011/files/2012/05/13...The paper also shows that the introduction of more representa-tive role

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