Dynamic Equality of Opportunity

Dynamic Equality of Opportunity

By JOHN E. ROEMER† and BURAK €UNVEREN‡

†Yale University ‡Yıldız Technical University

Final version received 3 March 2016.

What are the long-term effects of policies intended to equalize opportunities among different social classes

of children? To find out, we study the stationary states of an intergenerational model where adults are

either White or Blue collar employees. Both adults and the state invest in their children’s education. Our

analysis indicates that the major obstacle to equalizing opportunities in the long run is private educational

investment. Next we examine economies where only the state invests in education, motivated by the

Nordic experience. In a majority of these economies, no child lags behind regarding future prospects, a

theoretical result confirmed by simulations.

INTRODUCTION

While inequality of income is a ubiquitous phenomenon, it does not necessarily implyinjustice. The equality-of-opportunity ethic maintains that differences in income can bejust to the extent that they can be attributed to differences in effort, a factor for whichindividuals can be held responsible. In contrast, if income inequality is due to factors forwhich individuals should not be held responsible, then the inequality is unjust.

As a particular example, suppose that some children are expected to have lowerincomes as adults due to their lacking sufficient education. If those children are lesseducated because they had inferior educational resources, or home environments lesssupportive of education, then the resulting skill differences are not, at least obviously,due to a lack of effort, and may be unjust. The poor educational outcomes of somechildren, to the extent that they are caused by paucity of family resources, are not theresponsibility of the children.

Note how dynamics is an inherent part of the story in this example. Inequality ofincome among adults induces inequality of opportunity among children if poor parentscannot provide sufficient resources for the education of their children or if they lacksocial connections that rich parents possess. This will manifest itself in reproducingincome inequality when the children become adults whose children will, hence, also faceunequal opportunities. Therefore inequality of opportunity and inequality of incomepotentially feed each other over generations.

Equality of opportunity has emerged, in the last 20 years, as an attractive alternativeto welfarist approaches to social choice and welfare economics (see Roemer 1998, 2012;Fleurbaey 2008; Roemer and Trannoy 2016). As well as a burgeoning literature on thetheory of equal opportunity, there is now a growing empirical literature (see, forexample, Paes de Barro et al. 2009; Brunello and Checchi 2007; Peragine 2004). Boththeoretical and empirical literatures overwhelmingly take a static approach. In thetheoretical case, this means that the focus is on defining what the optimal opportunity-equalizing policy is at a moment in time, ignoring the dynamic issue that we raise here.

Here, we study an economy with successive generations. Each generation comprises acontinuum of households, and each household consists of a parent and child. Wepostulate that a parent is either a White collar professional or a Blue collar worker. Ateach generation, the state decides how much to invest in the education of children from

© 2016 The London School of Economics and Political Science. Published by Blackwell Publishing, 9600 Garsington Road,

Oxford OX4 2DQ, UK and 350 Main St, Malden, MA 02148, USA

Economica (2016)

doi:10.1111/ecca.12197

these two social backgrounds. The expenses of the state are financed by a linear incometax. Given the state’s policy, each parent privately invests in her own child’s education tomaximize a weighted sum of her own after-tax income and the expected after-tax incomeof her child. The objective of the state is to equalize opportunities for children, whichmeans investing in the education of children whose expected future prospects lag behinddue to factors for which no child can be held responsible (in our case, whether her parentis a White or Blue collar employee).

These decisions by the state and parents produce the next generation’s distribution ofWhite and Blue collar adults, who solve their optimization problems under the newcircumstances. The process continues indefinitely.

We want to understand the stationary states of this dynamic process.1 In particular,what do the stationary states look like that are optimal from the equal-opportunity pointof view? There are three possible kinds of stationary state: laissez-faire, moderate andideal. If the government collects no taxes and does not invest in education, then thesolution is laissez-faire. In a moderate solution, the government intervenes but the gapbetween the two types of children, when they enter the labour market, does not fullydisappear. In an ideal solution, no child lags behind in the labour market due to herparent’s type (Blue or White), and all children have identical future prospects in expectedterms. In this case, the state has been able fully to compensate Blue collar children for therelative disadvantage inherent in their background, whose nature will be made precisebelow.

We conduct both simulations and theoretical analysis. In our simulations, laissez-faire turns out to be the most frequent kind of solution, and the least frequent solution isthe ideal one. The theoretical analysis explains this regularity. We show that for a largeset of parameter vectors, White collar parents react to state investment in Blue collarchildren by increasing private investment in their own children, undoing the effect of(well-intended) state policy. Therefore the state, in a plurality of cases, finds it optimalnot to invest in Blue collar children. If the impact of education on productivity issufficiently high, then the solution is moderate, meaning that the gap between children’sopportunities is reduced, but still persists. The ideal solution is observed only when theefficacy of investment in education is low.

These results point to a perpetual inequality among children due to their parentalbackgrounds when private supplements to public education are available, even if thestate’s objective is providing equality of opportunity for everyone. Thus, inspired by theNordic experience in education, where there is negligible privately funded education, weask what happens if there is only public investment in education. In this case, the idealsolution holds in a clear majority in all solutions in the simulations. We also prove thatthe solution should indeed be ideal, which ensures that no child’s chances are inhibited byhis/her parent’s type except when the impact of education on future outcomes is very low.

We proceed by specifying the model, explaining the optimization problem, reportingour simulations, and deriving the theoretical results.

I. THE ENVIRONMENT

We consider an infinite horizon dynamic economy with successive non-overlappinggenerations. A generation comprises a continuum of households. Each householdconsists of a parent and a child. Both parents and the state invest in children’s education.Children from White collar backgrounds possess an advantage over children from Bluecollar backgrounds in the labour market.

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© 2016 The London School of Economics and Political Science

2 ECONOMICA [MAY

In formal terms, each parent is either a White collar (W) professional or a Blue collar(B) employee. The pre-tax income of a parent with a type J occupation is yJ, forJ 2 {B, W}. The income yJ is a function of the education that the type J parent hadduring her childhood.

The sum of private and public investments in a child of type J is

zJ ¼ iJ þ sJ

for any child whose parent is of type J. Here iJ ≥ 0 and sJ ≥ 0 are the private and publicinvestments, respectively, in the child’s education. Public investment in education isfinanced by an affine income tax at rate t. The budget constraint of the state will bediscussed below.

Write ξ = (t, sB, sW, iB, iW) for the set of all endogenous variables of the model.Consider a child whose parent is of type J. We hypothesize that the child’s income in heradulthood will be either

yBðnÞ ¼ hB þ bzJ

if the child becomes a B worker, or

yWðnÞ ¼ hW þ bzJ

if the child becomes a W professional. Note that the vector of endogenous variablesξ = (t, sB, sW, iB, iW) gives us the total investment in each type of child, zJ. Theparameters (hB, hW) are exogenously given basic-level wages for both types of employee.They correspond to the income of each type of employee in the case of no investment ineducation. We assume hB < hW. Moreover, b is the marginal impact of investment ineducation on income with certainty.

Note that two adults of the same type (e.g. two White collar professionals) may havedifferent incomes. This happens when the past investment in the education of these twoadults differed. Thus not all adults of the same type earn the same income.

Whether a child will become a Blue or a White collar employee in adulthood is astochastic event that is determined by the endogenous vector ξ. More formally, theprobability that a child whose parent has a type J job will become aW adult is

pJðnÞ; J 2 fB;Wg:

A parametric form of pJ will be assumed presently. Each J parent’s standard of livingis

uJðnj�nÞ ¼ yJð�nÞ ð1� tÞ � iJ;

where ξ is the present generation’s endogenous variable vector and �n is the pastgeneration’s endogenous variable vector. Recall that the present adult’s income yJ is afunction of the past generation’s �n . Let ξ* denote the future generation’s endogenousvariables. Hence, by definition, the expected standard of living of the child is

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2016] DYNAMIC EQUALITY OF OPPORTUNITY 3

EJðn�jnÞ ¼ pJðnÞ uWðn�jnÞ þ ð1� pJðnÞÞ uBðn�jnÞ:

Each J parent solves

maxiJ

uJðnj�nÞ þ / EJðn�jnÞ;ð1Þ

where φ represents the welfare weight on the child’s standard of living from the parent’sviewpoint. (To do this, the parent must know ξ*; this will be the case in the stationarystate.) Parents care about both themselves and their children. Note that we take theexpectation of the child’s material payoff (i.e. standard of living) because whether she willbe a Blue collar or a White collar employee is uncertain, and may be interpreted as due toluck or to effort in the job market.

Today’s optimal iJ is typically a function of future ξ*. (That is, a parent mustcontemplate her child’s future investment in her child, to determine the child’s expectedstandard of living.) However, optimal iJ today does not depend on past �n .

Lemma 1 Optimal iJ of equation (1) does not depend on past endogenous �n .

Proof Define

UJ ¼ uJðnj�nÞ þ / EJðn�jnÞ:

It is easy to see that �n does not appear in

dUJ

diJ¼ �1þ /

dEJðn�jnÞdiJ

;ð2Þ

establishing the claim.Lemma 1, though obvious, is crucial, because it ensures that the number of different

income levels does not increase over time but stays bounded. In fact, at any givengeneration, there can be at most four different income levels.2 Since optimal iJ does notdepend on past �n , it follows that optimal iJ is identical among all type J families,although two type J adults’ incomes may be different, as can be seen readily fromequation (2). As a consequence, zJ = iJ + sJ is identical among all type J families. Inother words, both zB and zW are well-defined at each generation. Now recall that theincome of a B worker is either hB + bzB or hB + bzW. Thus there are at most two levels ofincome of B workers. The same argument applies toW professionals, so there are at mostfour different income levels.

Given that zJ is identical among all type J families, we postulate that

pWðnÞ ¼ 1

1þ e�xand pBðnÞ ¼

a

1þ ex;ð3Þ

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where

x ¼ bzW � zBhW � hB

; a 2 ð0; 1Þ and b[ 0:

Note that the probability of becoming a White collar professional depends onDz = zW � zB for all children—the difference in investment in the education of the twotypes of child. The role of Dz in pJ is to capture the competitive edge that is providedby education. For instance, think of education as a signalling device in the labourmarket where young potential employees compete for White collar positions.Employers deem better education as a signal of competence for the tasks required fromWhite collar professionals. However, the strength of education as a signalling device isnegatively related to Dh = hW � hB. That is, more resource must be invested ineducation to become a White collar employee as the relative advantage of being aWhite collar employee increases. The parameter b > 0 represents the efficacy ofinvestment in education.

The fact that a < 1 models the idea that Blue collar children are disadvantagedcompared to their White collar counterparts. This can be interpreted as due tonetwork effects. Bewley (1999) estimates that 30–60% of jobs were found throughfriends or relatives. Corak (2013) reports findings that show that the sons of veryhigh earning fathers take first jobs in their father’s firm much more frequently thando the sons of other fathers. In the present setting, our interpretation is that Whitecollar parents help their children to find White collar jobs using their high-levelsocial connections. A Blue collar parent lacks this advantageous network. Chettyet al. (2015) report that in an experiment in which poor children moved to betterneighbourhoods, their income increased by 31% compared to children who did notmove.

As for the population dynamics, write f sJ for the fraction of type J employees ingeneration s. Of course, f sW þ f sB ¼ 1. Normalize the population mass to unity. The percapita mean income in generation s is

lðnÞ ¼ f sWhW þ f sBhB þ bðf s�1W zW þ f s�1

B zBÞ;

where ξ represents the endogenous variables in generation s - 1. The budget constraint ofthe state at generation s is

t lðnÞ ¼ f sWsW þ f sBsB:

Population fractions evolve according to

f sþ1W ¼ pBf

sB þ pWf sW;

while fsþ1B can be deduced from the identityf sþ1

B ¼ 1� f sþ1W . In the stationary state,

f sJ ¼ f sþ1J for each J 2 fB;Wg:

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Thus in the stationary state we have

f �W ¼ pBf�B þ pWf �W

or

f �Wð1� pWÞ ¼ pBf�B ¼ pBð1� f �WÞ

or

f �W ¼ að1� f �WÞ;

which implies

f �W ¼ a

1þ a; f �B ¼ 1

1þ a:

In other words, the two types of job occur in fixed proportions in all stationary states.

II. STATIONARY STATES AND EQUALITY OF OPPORTUNITY POLICY

Stationary states

A concern in the long run suggests that we seek a policy whose induced stationary state isbest from the viewpoint of the social objective—in our case, equalizing opportunities.First we define a stationary state, and then we define what ‘equalizing opportunities’means.

A stationary state is a state policy (t, sW, sB) and private investment decisions of thehouseholds (iW, iB) that are constant over time and induce a stationary distribution ofjob types and income. In particular, in a stationary state, investment decisions taken bythe households are constant over time, when each expects the policy to remain fixed.

In formal terms, a stationary state is a vector n� ¼ ðt�; s�B; s�W; i�B; i�WÞ such that

t� lðn�Þ ¼ f �Ws�W þ f �Bs�B

and i�J is a solution in iJ to

maxiJ

uJðn0jn�Þ þ / EJðn0jn�Þð4Þ

subject to

n0 ¼ ðt�; s�B; s�W; iB; i�WÞ if J ¼ B;

ðt�; s�B; s�W; i�B; iWÞ if J ¼ W;

�

for each J.

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Of course, stationary-state analysis cannot be used to study a growing economy. Ifb ≥ 1, then any arbitrarily high ðs�W; s�BÞ is financially feasible given that t* 2 (0, 1) ischosen accordingly. Therefore we assume b 2 (0, 1) to ensure that no tax rate canfinance arbitrarily high expenditures by the state. Without such a restriction on b,stationary-state analysis cannot be conducted, and we would need an endogenous growthmodel to analyse balanced growth paths. This would, of course, take us far from ourobjective—understanding the dynamic trade-offs involved in the equality of opportunityparadigm.

Let Ξ denote the set of all stationary states ξ*. Observe that any stationary state ξ*,by definition, is both incentive-compatible from the standpoint of households andfinancially feasible from the standpoint of the state. Hence it only remains to specify thesocial objective. This will determine which stationary state should be chosen from allpossible elements of Ξ.

Equality of opportunity

In our model, the clear circumstance demanding compensation is the parental status ofthe parent as B or W, and hence also the parental investments in the education of theirchildren. However, one must also specify what lies behind the stochastic nature ofoccupational assignment—namely, the probabilities pW and pB. Is the fact that somechildren of a given type become W adults and some become B adults due to differentialeffort, or to other circumstances that are unnamed?

We take the position that all accomplishments and behaviour of children be classifiedas due to circumstances: that is, children should be held responsible for nothing until an‘age of consent’ is reached (perhaps 14, 16 or 18 years of age). This does not mean thatone should refrain from punishing and rewarding children for kinds of behaviour,actions that can instil a sense of responsibility that will help the individual as an adult.Therefore it would be inconsistent for us to interpret the stochastic element in the modelas due to differential effort of children. Instead, we think of adult incomes as permanentincomes, and interpret the stochastic element as due to differential effort among adults.Accordingly, we define the equality of opportunity (EOp) policy as one that seeks toerase the differential expected incomes of adults coming from different socialbackgrounds. We use Van de gaer’s (1993) version of the EOp objective, which can besummarized as the minimum of means across types. The socially optimal stationary stateis therefore the solution of the programme

maxn�2N min½ �EBðn�Þ; �EWðn�Þ�;ð5Þ

where EJðn�jn�Þ ¼ �EJðn�Þ for J = B, W.We note in passing what is a common feature in thinking about EOp, that the effort

of one person may well become a circumstance for another. Thus an adult may havebecome a W adult through hard work, but her high status becomes a circumstance forher child. Adults have no right, according to this ethic, to pass on special advantage totheir child by virtue of their own effort. A similar remark applies, for example, toinheritance. It is not inconsistent to view some wealth as justly acquired yet forbid orhighly restrict transferring it to one’s children.3

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Types of optimal stationary state

At any solution ξ* of the EOp programme, either �EWðn�Þ ¼ �EBðn�Þ or not. In the formercase, all children are equally well off in expected terms, independent of their parent’stype. We call this an ideal solution. Otherwise, EOp is incomplete at the optimum. If thisinequality is associated with t* = 0, then we call the solution laissez-faire. When thesolution is laissez-faire, the state does nothing to support the disadvantaged children. Ofcourse, inequality between �EWðn�Þ and �EBðn�Þ can coexist with t* > 0. In this case, wecall the solution moderate. At a moderate solution, the state actively intervenes butcannot fully eradicate the inequality between children from different backgrounds.

In sum, the three different possible regimes are listed in Table 1.

III. ANALYSIS

In this section we begin our analysis by conducting numerical simulations. Certainpatterns will emerge in simulating and calibrating the model. Theoretical explanationswill follow.

Numerical analysis

The first question that we address is the distribution of the three cases as defined inTable 1 in the space of parameters. We first use numerical simulation to study thisquestion. The vector of exogenous variables is e = (a, b, φ, b, hW, hB). We ransimulations by drawing 1000 random samples of e, calculating the solution to the EOpprogramme as defined in (5) for each case, and finally counting the number of laissez-faire, moderate and ideal regimes in these 1000 solutions.

In the first numerical simulation, the support of the parameters is set to be relativelywide. To be specific, random samples are drawn as follows: (a, φ, b) 2 (0, 1)3,hW 2 (0, 100), hB 2 (0, hW) and b 2 (0, 50). All parameters are assumed to bedistributed uniformly on their supports. We obtain the distribution of solutions reportedin Table 2.

Laissez-faire policy has the highest frequency, and is followed by the moderateregime. Ideal solutions comprise the lowest frequency. Moreover, as one would expect,�EWðn�Þ � �EBðn�Þ in all cases. Nonetheless, �EWðn�Þ ¼ �EBðn�Þ occurs in only 29.5% of

TABLE 1ALL POSSIBLE SOLUTIONS TO PROGRAMME (5)

Laissez-faire Moderate Ideal

t = 0 > 0 > 0�EWðn*Þ � �EBðn*Þ�� > 0 > 0 = 0

TABLE 2PERCENTAGE SOLUTIONS OF THE EOP PROGRAMME OF THREE TYPES


37.7 32.8 29.5

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cases. Therefore in a clear majority of the cases �EWðn�Þ[ �EBðn�Þ: children of Whitecollar parents are better off than the children of Blue collar parents in 70.5% of cases.

We also report the averages of certain variables of interest in this simulation, inTable 3. These variables are the tax rate, the probability of becoming a W professional,the ratio of expected future incomes, and how much is invested in education, as aproportion of income, according to type.

It is noteworthy that the averages of t at moderate and ideal solutions are similar. Incontrast, laissez-faire and moderate solutions exhibit a similarity in average pW. Ofcourse, pB = a(1 - pW) can be used to retrieve the average value of pB at each solution.The average ratios of �EW to �EB at laissez-faire and moderate solutions are around 2,while the same ratio is, by definition, exactly 1 at any ideal solution.

In order to see whether there is an observable regularity, we also ran a simulationthat admits a visual, geometric relation between parameters and the solution to the EOpprogramme; see Figure 1.

The random samples for the simulations in Figure 1 are drawn as follows. First, wefix a = 0.56, hW = 100, hB = 58 by calibrating these parameters to the US data. Ourcalibration method, and the data are explained in the Appendix. Since b and b are hardlyobserved in real life, (b, b) 2 (0, 50) 9 (0, 1) are randomly chosen. As for φ, we impose0.4 and 0.9, respectively. Our aim is to see the impact of φ on the results as clearly aspossible. The number of the random samples is 1000 for each part of the figure.

We see the following regularities. If b is high, then we observe no ideal solutions, i.e.�EW � �EB [ 0 is always true. If, however, b is high, then there is no laissez-faire solution,that is, optimal EOp taxation is always positive. Finally, comparing the two parts ofFigure 1 shows that the number of ideal solutions decreases as φ increases.

Finally, we investigate the possible conflict between opportunity-egalitarian andutilitarian ethics. The utilitarian would choose the stationary state that maximizesincome per capita, which is the same thing as total expected income per capita in ourmodel. Critics of EOp in educational policy often say that if taken too far, the average

TABLE 3

AVERAGES OF CERTAIN ENDOGENOUS VARIABLES

t pW �EW= �EB fWzW/l fBzB/l

Laissez-faire 0 0.85 2.18 0.15 0Moderate 0.13 0.83 1.83 0.08 0.12Ideal 0.11 0.36 1 0.04 0.17

1.0

0.8

0.6

0.4

0.2

1.0

Laissez-faireModerateIdeal

0.8

0.6

0.4

0.2

10 20 30 40 50b

10 20 30 40 50b

β β

FIGURE 1. Distribution of all solutions when φ = 0.4 (left) and φ = 0.9 (right).

Economica



skills of the population will suffer. Some say that ambitious parents will cease investingso much in their children if they see the effect being undone by state subsidies to thedisadvantaged. To study this, we chose the vectors where the EOp solution was ideal.There are 295 parameter vectors that satisfy this criterion. We then restricted the tax rateto be zero for these economies, and calculated the stationary state—in other words, thatstationary state in which only private investment in education occurs. We denote the zerotax rate stationary states as ones of ‘mandatory laissez-faire’. Table 4 presents thecomparison of the ideal EOp stationary state and the mandatory-laissez-faire stationarystate.

First, we see that there is no conflict between utilitarian and equal-opportunity goalshere: that is, average income is higher in the ideal EOp solutions than in the mandatorylaissez-faire solutions. Second, total investment in education is much higher in the idealEOp solutions. It is true that W parents are dissuaded from investing in their children inthe ideal EOp solution (crowding out), but the state actually invests more in W childrenin the ideal EOp solution than the W parents invest in them in the mandatory laissez-faire solution. Unsurprisingly,W children fare somewhat worse in the ideal EOp solutionthan in the mandatory laissez-faire solution, but the improvement in the expectedincomes of the B children more than compensates for this, from the social (i.e. utilitarian)viewpoint.

We next study how many of these regularities we can deduce from the model.

Theoretical analysis

In this subsection we attempt to explain the regularities observed in the numericalsimulations. Let us first focus on the pervasiveness of �EWðn�Þ[ �EBðn�Þ. We have thefollowing:

Theorem 2 Let ξ* solve the EOp programme. If i�W [ 0, then �EWðn�Þ[ �EBðn�Þ.

Proof Suppose that n� ¼ ðt�; s�B; s�W; i�B; i�WÞ solves the EOp programme, and i�W [ 0.

The first step of the proof demonstrates that pW ≥ 1/2.

To see this, consider the first-order optimality condition for i�W [ 0

dUW

diW¼ �1þ /

d �EWðn�ÞdiW

¼ 0;

TABLE 4

STATISTICS FOR ECONOMIES WHERE IDEAL EOP IS OPTIMAL, BUT RESTRICTED TO BEING

LAISSEZ-FAIRE

iW sW iB sB l EW EB

Ideal EOp 0 3.85 0 5.4 30.93 25.52 25.52Mandatory laissez-faire 1.95 0 0 0 28.06 37.15 23.46

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which is equivalent to

�1þ / bpWð1� pWÞ DuDh

þ ð1� t�Þb� �

¼ 0;ð6Þ

where Dh = hW - hB, and all relevant variables are, of course, evaluated at ξ*. However,Du = uW - uB > 0, for otherwise,


þ ð1� t�Þb� �

� � 1þ /ð1� t�Þb\0:

The second-order condition of theW parent’s optimization problem is

/b2ð1� 2pWÞpWð1� pWÞ Du

Dh2� 0:

Since Du > 0, it follows that 1 � 2pW ≤ 0, as claimed. However,

�EWðn�Þ � �EBðn�Þ ¼ ðpW � að1� pWÞÞDuþ bDzð1� t�Þ:

Hence if Du > 0, pW > a/(1 + a) and Dz = zW - zB > 0, then �EWðn�Þ � �EBðn�Þ[ 0.But we showed that i�W [ 0 implies pW ≥ 1/2 and Du > 0. (See the definition of pW to seethat pW ≥ 1/2 implies zW � zB > 0.) Note that the maximum value of a/(1 + a) is alwaysless than 1/2. Conclude that i�W [ 0 implies �EWðn�Þ[ �EBðn�Þ.

Theorem 2 explains why �EWðn�Þ[ �EBðn�Þ is such a common case. An interiorsolution to the W parent’s maximization problem is sufficient for incomplete EOp:�EWðn�Þ[ �EBðn�Þ.

We next show that if b is sufficiently high, then �EWðn�Þ[ �EBðn�Þ (as in Figure 1).

Theorem 3 Let ξ* solve the EOp programme. If b is sufficiently high, then�EWðn�Þ[ �EBðn�Þ.

Proof Suppose that �EWðn�Þ ¼ �EBðn�Þ, where ξ* is the solution to the EOpprogramme. Then from Theorem 2, i�W ¼ 0, thus

�EWðn�Þ � �EBðn�Þ ¼ pW � að1� pWÞð Þ Dh ð1� t�Þ þ b Dz ð1� t�Þ:ð7Þ

It follows that

að1� pWÞ � pW ¼ bDzDh

:ð8Þ

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But the definition of pW implies

DzDh

¼ 1

blog

pW1� pW

� �:ð9Þ

Substituting Dz/Dh from equation (9) into equation (8) gives

að1� pWÞ � pW � bblog

pW1� pW

� �¼ 0:

Deduce that pW ? a/(1 + a) as b ? ∞. As a consequence, dUW/diW ? ∞ asb ? ∞ since

dUW

diW¼ �1þ /ðbpWð1� pWÞ þ bÞð1� t�Þ:

Therefore dUW/diW > 0 is implied by sufficiently high b, contradicting i�W ¼ 0.Large b means that the impact of investment in education on the probability of

becoming a W adult is high. In such a case, W parents find it rational to invest positiveamounts. Hence it follows that an ideal solution to the EOp programme cannot occur if bis high enough.

Note that Theorem 2 can be rephrased as follows: i�W ¼ 0 is a necessary condition forcomplete EOp among children from different backgrounds. Under what conditions canwe be sure that i�W ¼ 0? The answer turns out to be pertinent to another regularity in thesimulations.

Proposition 4 Suppose that φ((b/4) + b) < 1. Then i�W ¼ 0.

Proof Recall that the first-order optimality condition ofW parents is


þ ð1� t�Þb� �

� 0 ðwith equality if i�W [ 0Þ:

Note that pW(1 - pW) ≤ 1/4 and (1 - t*)b ≤ 1. Now we will show that i�B ¼ 0 at anyξ*. Consider the problem of the B parent whose first-order condition satisfies

dUB

diB¼ �1þ / abpWð1� pWÞ Du

Dhþ ð1� t�Þb

� �\� 1

þ / bpWð1� pWÞ DuDh

þ ð1� t�Þb� �

¼ dUW

diW� 0:

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12 ECONOMICA [MAY

Deduce that i�B ¼ 0. Therefore Du ¼ Dhð1� t�Þ � ði�W � i�BÞ�Dh . As a consequence,


þ ð1� t�Þb� �

� � 1þ /b

4þ b

� �\0;

where the last inequality follows from the hypothesis of the proposition. We concludethat i�W ¼ 0. □

Hence a small value of φ ensures zero private investment by W parents, which isintuitively unsurprising. Our first theorem states that this is a necessary condition for ξ*to be an ideal solution. Now we can explain why the number of ideal solutions in thesimulations decreases when φ increases, as can be observed easily from Figure 1. When φis very low, W parents do not care much about their children and make no investment inthem. Therefore private investment in education cannot impede state policies designed toimplement �EWðn�Þ ¼ �EBðn�Þ.

Another clear regularity in the simulations is that there is no exception to theinequality �EWðn�Þ� �EBðn�Þ. The next theorem shows why.

Theorem 5 Let ξ* solve the EOp programme. Then generically �EWðn�Þ � �EBðn�Þ.

Proof We need to consider only the cases in which i�W ¼ 0, because when i�W [ 0,�EWðn�Þ[ �EBðn�Þ has already been proved.

Suppose that the claim were false. Consider a solution at which �EWðn�Þ\ �EBðn�Þ. Weknow i�B ¼ 0 as we proved above, and by hypothesis, i�W ¼ 0. Thus Du = Dh(1 - t*). Nowsuppose that s�B [ 0. Then foregoing a small part of s�B to finance a small increase in s�Wwhile keeping t* constant would generically not affect private investment decisions. But itwould increase

�EWðn�Þ ¼ pWuW þ ð1� pWÞuB ¼ pW Duþ uB;

because Du is positive, and pW is monotonically increasing in s�W and decreasing in s�B.But then increasing the expected standard of living of the worse-off would be feasible.Conclude that s�B [ 0 cannot happen if �EWðn�Þ\ �EBðn�Þ. Thereforen� ¼ ðt�; s�B; s�W; i�B; i

�WÞ ¼ ðt�; 0; s�W; 0; 0Þ, which ensures that pW > 1/2 > a/(1 + a). But

pW > a/(1 + a) implies

�EWðn�Þ � �EBðn�Þ ¼ ðpW � að1� pWÞÞ Duþ bzWð1� t�Þ[ 0:

Next we address what is arguably the most significant regularity in the simulations.We see that if b is high enough, then the solution is never laissez-faire in the simulations.That is, if b is sufficiently high, then t* > 0.

Theorem 6 If b is sufficiently large, then t* > 0 at any solution to the EOp programme,ξ*. More specifically:

(i) b[ hBahWþhB

implies t* > 0 when i�W [ 0;

Economica



(ii) b[ ahWþð1�aÞhBahWþhB

implies t* > 0 when i�W ¼ 0.

Proof Let ξ = (t, sB, sW, iB, iW) 2 Ξ be an arbitrary stationary state. We will showthat if b is sufficiently large and t = 0 at ξ, then ξ cannot solve the EOp programmebecause a very small increase in t increases the value of �EB. This is sufficient to prove theclaim, since �EB\ �EW at any stationary state ξ if t = 0.

In particular, holding sW constant, the impact of an infinitesimal change in t on �EB is

d �EB

dt¼ @ �EB

@tþ @ �EB

@sB

@sB@t

þ @ �EB

@iW

@iW@t

:ð10Þ

We compute

@ �EB

@t¼ �að1� pWÞ Dh� ðhB þ bzBÞ;ð11Þ

@ �EB

@sB¼ abð1� pWÞpW

DuDh

þ bð1� tÞ;ð12Þ

@ �EB

@iW¼ �abð1� pWÞpW

DuDh

� að1� pWÞ;ð13Þ

where Dh = hW - hB.Now we derive @ sB/ @ t and @ iW/ @ t. Given t, the pair (sB, iW) is a solution to the

non-linear equation system

tðahW þ hB þ bðazW þ zBÞÞ � asW � sB ¼ 0;


þ ð1� tÞb� �

¼ 0:

The first equation is the budget constraint, and the second equation is the optimalitycondition of W parents, which stipulates an interior solution. The case of boundarysolution, i.e. i�W ¼ 0, is addressed at the end of the proof. Now write this equation systemas

FðsB; iW; tÞ ¼ 0:

By the implicit function theorem,

@sB@t@iW@t

" #¼ � DsBF

DiWF

� ��1

�DtF;

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14 ECONOMICA [MAY

whereD is the differentiation operator. Note that

DsBF

DiWF

" #¼

bt� 1 atb

�/pWð1� pWÞ Du ð1� 2pWÞ bDh

� 2/pWð1� pWÞ Du ð1� 2pWÞ b

Dh

� 2 � bDh

�" #

and

DtF ¼ lð1þ aÞ�/ðbpWð1� pWÞ þ bÞ� �

:

Since t = 0 it follows that

DsBF

DiWF

" #�1

¼ 1

/pWð1� pWÞ Du ð1� 2pWÞ bDh

� 2 � bDh

�/pWð1� pWÞ Du ð1� 2pWÞ b

Dh

� 2 � bDh

�0

/pWð1� pWÞ Du ð1� 2pWÞ bDh

� 2 �1

24

35;

which implies that

@sB@t

¼ lð1þ aÞ; @iW@t

¼ �Du ð1� 2pWÞ b

Dh

� 2lð1þ aÞ þ b

/pWð1�pWÞ þ b

bDh � b

Dh

� 2Du ð1� 2pWÞ

ð14Þ

at a stationary state ξ such that t = 0. Using equations (11)–(14) to expandequation (10), we obtain

d �EB

dt¼� að1� pWÞðhW � hBÞ � hB þ abð1� pWÞpW

DuDh

þ b

� �lð1þ aÞ

þ abð1� pWÞpWDuDh

þ að1� pWÞ� �

Du ð1� 2pWÞ bDh

� 2lð1þ aÞ þ b

/pWð1�pWÞ þ b

bDh � b

Dh


:

Economica



We deduce that

d �EB

dt¼� að1� pWÞ Dh� hB þ abð1� pWÞpW

DuDh

þ b

� �lð1þ aÞ

þ að1� pWÞ 1þ pWbDuDh

� � b/pWð1�pWÞ þ b

bDh � b

Dh


�lð1þ aÞ 1�bDh

bDh � b

Dh


!!

¼ �að1� pWÞ Dh� hB þ ðb� að1� pWÞÞ lð1þ aÞ

þ að1� pWÞ 1þ pWbDuDh

� � b/pWð1�pWÞ þ lð1þ aÞ b

Dh þ b

bDh � b

Dh


:

Since pW > � (1 � 2pW), we have

d �EB

dt¼� að1� pWÞ Dh� hB þ ðb� að1� pWÞÞ lð1þ aÞ þ að1� pWÞðDhþ b

þ lð1þ aÞÞ[ � hB þ blð1þ aÞ� � hB þ bðahW þ hBÞ;

which implies

d �EB

dt[ 0 if b[

hBahW þ hB

and i�W [ 0:

As for the case in which i�W ¼ 0, one can show that

d �EB

dt¼ �að1� pWÞðhW � hBÞ � hB þ abð1� pWÞpW

DuDh

þ b

� �lð1þ aÞ

� � að1� pWÞðhW � hBÞ � hB þ blð1þ aÞ� � aðhW � hBÞ � hB þ blð1þ aÞ;

which implies

d �EB

dt[ 0 if b[

ahW þ ð1� aÞhBahW þ hB

: h

Empirical discussionWhat can be inferred about reality from this theoretical discussion? We argue that—

when confronted with data—our model points to the moderate solution as a more likelyoutcome if the policymakers possess an equal-opportunity ethic.

First, let us see why the ideal solution is a less likely outcome. The ideal solution canoccur only if W parents do not privately invest in their children (Theorem 2).Nonetheless, relatively affluent and well-educated parents typically invest in their

Economica


16 ECONOMICA [MAY

children’s education in real life. This basic empirical observation means that the idealsolution will not occur; it can occur only if the government forbids private investment ineducation or a social norm prevents it. We will discuss provision of education only inpublic schools in the next section.

Second, laissez-faire is a less likely outcome in real life. As we will see now, relativelylow values of b discourage the laissez-faire solution by creating ample incentive for activetaxation. Here we invoke Theorem 6 and some basic data. Recall that

b[1

ahW=hB þ 1

ensures active taxation for EOp, t* > 0 (Theorem 6). Using OECD data, we calibratethis threshold level of b given by 1/(ahW/hB + 1) for selected countries. Our method ofcalibration and the data are explained in the Appendix.

As can be seen in Table 5, the average threshold of b is 0.53: b > 0.53 implies that, onaverage, active taxation is optimal for policymakers whose objective is EOp. The highestcalibrated value of the threshold of b is 0.6 for Sweden, Canada and Japan.

But is the actual value of b greater than 0.6? Although we cannot derive a precisevalue for b, a rough picture can be obtained as follows. According to Psacharopoulosand Patrinos (2004), private returns to education in OECD countries vary between 11%and 13%. Therefore, in the current setting, the marginal impact of private investment onexpected income is

d �EW

diW� 1:12:

In words, $1 investment in education implies $1.12 increase in expected income.However, this rate of return is not risk-free. The uncertainty in this rate of change stemsfrom the fact that whether a child will be aW or B collar worker is a random event in ourmodel. The increase in income due to education that does not depend on this riskyoutcome is represented by the parameter b. Since, however, education is typically not ashort-run investment, its return presumably involves low risk, which would be the casewith high b. Hence we deem that b > 0.6 is plausible. If this is the case, active taxation isoptimal in the EOp programme for all the countries in Table 5. Of course, further dataand deeper empirical analysis would contribute to the discussion because we cannotdetermine the exact value of b. We believe that it is not far-fetched to think that b is closeto 1 if education is indeed a relatively safe investment of parents for their children in thelong run.

In conclusion, the moderate solution seems to be a more likely outcome if theobjective of the policymakers is to equalize opportunities when our model is confrontedwith data.

TABLE 5

CALIBRATION OF THE THRESHOLD OF b

Average Sweden Norway Canada UK USA Germany France Japan

Threshold of b 0.53 0.6 0.45 0.6 0.5 0.5 0.48 0.52 0.6

Economica



IV. ONLY PUBLIC SCHOOLS

Our analysis suggests that private investment in education is a major obstacle toobtaining EOp among children from different backgrounds. In particular, we proved thati�W [ 0 implies that �EWðn�Þ[ �EBðn�Þ. Moreover, sufficiently high b also implies�EWðn�Þ[ �EBðn�Þ. Finally, the number of cases in which �EWðn�Þ ¼ �EBðn�Þ decreases as φincreases in our simulations. This numerical result is related to the theoretical fact thatthat if φ is low enough, then i�W ¼ 0. Nevertheless, we find low φ an empiricallyimplausible condition: parents do value their children’s welfare quite highly in real life.

Therefore we now posit that the state forbids private investment in education, or thata social ethos prevents it, which may be the historical explanation of the Nordic practice.Now we seek a solution to the following programme in ξ*:

maxn� min½ �EBðn�Þ; �EWðn�Þ�ð15Þ

subject to

t�lðn�Þ ¼ f�Ws�W þ f�Bs�B; i

�J ¼ 0; J ¼ B;W; n� 2 ½0; 1� R4þ:

Let us call (15) the Nordic EOp programme. The interpretation is that the state aimsto equalize opportunities by investing in education in an economy where, for somereason, we constrain iW = iB = 0.

We can calculate the distribution of laissez-faire, moderate and ideal solutions to theNordic EOp programme by means of numerical simulations. Random samples are againdrawn as follows: (a, φ, b) 2 (0, 1)3 and b 2 (0, 50), hW 2 (0, 100) and hB 2 (0, hW),while all parameters are assumed to be distributed uniformly on these supports. Wereport the distribution of regimes in Table 6.

When the incentive compatibility constraints are dropped, we see a dramatic changein the results. The ideal solution comprises 80.4% of all solutions. The cases in which�EWðn�Þ[ �EBðn�Þ drop to 19.6% when there is investment in education only by the state.

To understand the reason behind this change in the distribution of solutions, let usrun another simulation that can be visualized. We fix a = 0.56, φ = 0.9, hB = 58,hW = 100. These are the calibrated values of these parameters to the US data. We draw1000 samples of b 2 (0, 1) and b 2 (0, 50).

In an overwhelming majority of the cases, the solution to the Nordic EOpprogramme is ideal (as can be seen in Figure 2). More specifically, we observe that if band/or b are sufficiently high, then the solution induces �EWðn�Þ ¼ �EBðn�Þ. Indeed, thisvisual regularity in our simulations is verified by a theoretical result.

TABLE 6

PERCENTAGES OF ALL SOLUTIONS WITHOUT PRIVATE INVESTMENT, NORDIC PROGRAMME


10.2 9.4 80.4

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18 ECONOMICA [MAY

Theorem 7 If b or b is sufficiently large, then the solution to the Nordic EOp ξ* is ideal:�EWðn�Þ ¼ �EBðn�Þ.

Proof We first claim that �EWðn�Þ� �EBðn�Þ at any solution ξ* to the Nordic EOpprogramme. Assume that the claim is false. Then children of W adults are worse off atsome ξ*. There are two cases: s�B [ 0 or s�B ¼ 0. In the former case, a small decrease in s�Bto increase s�W while keeping t* constant would increase

�EWðn�Þ ¼ pWuW þ ð1� pWÞuB ¼ pW Duþ uB

since Du = (hW - hB)(1 - t*) > 0 and pW is monotonically increasing in s�W and decreasingin s�B. But then increasing the expected standard of living of the worse-off would befeasible. Conclude that s�B [ 0 cannot happen if �EWðn�Þ\ �EBðn�Þ. Therefore s�B ¼ 0,which implies

n� ¼ ðt�; s�B; s�W; i�B; i�WÞ ¼ ðt�; 0; s�W; 0; 0Þ;

so pW > 1/2. But 1/2 > a/(1 + a) implies pW > a/(1 + a). Therefore we conclude that�EWðn�Þ[ �EBðn�Þ, as we proved earlier. This contradiction demonstrates the claim.

Now assume that there is a solution ξ* such that �EWðn�Þ[ �EBðn�Þ. Consider theimpact of a small change in s�B on �EB while keeping s�W fixed:

d �EB

dsB¼ @ �EB

@sBþ @ �EB

@t

@t

@sB;

where

@ �EB

@sB¼ abð1� pWÞpW

DuDh

þ bð1� tÞ;

1.0

0.8

0.6

0.4

0.2

Laissez-faireModerateIdeal

10 20 30 40 50b

β

FIGURE 2. Distribution of all solutions when some variables are fixed.

Economica



@ �EB

@t¼ �að1� pWÞ Dh� ðhB þ bzBÞ;

@t

@sB¼ 1� bt

l:

Of course, @t/ @sB is derived invoking the budget constraint. Hence

d �EB

dsB¼ abð1� pWÞpW

DuDh

þ bð1� tÞ � ðað1� pWÞ Dh� ðhB þ bs�BÞÞ1� bt

l

[ abð1� pWÞpWDuDh

þ bð1� tÞ � ð1� btÞ� abð1� pWÞpW � ð1� bÞ:

Obviously, if b or b is sufficiently high then ab(1 � pW)pW � (1 � b) > 0, whichimplies that d �EB=dsB [ 0. Therefore ξ* such that �EWðn�Þ[ �EBðn�Þ cannot be a solutionto the Nordic EOp if b or b is large enough. We conclude that if b or b is large enough,then �EWðn�Þ ¼ �EBðn�Þ at any solution to the Nordic EOp.

V. CRITIQUE AND CONCLUSION

Critique may be made of our assumption that the state differentiates its expenditures bythe type of parental occupation rather than by the income of the parent. It is theoreticallypossible that some Blue collar workers earn more than some White collar workers, ifmuch more is invested in the former’s education than in the latter’s. Would it be moredefensible, then, for the state to target children according to parental income rather thanoccupation?

Our response is that we must be interested in each child’s expected standard of living,not how rich her parents were per se. And the expected standard of living of all childrenfrom Blue collar families is the same regardless of how rich their parents are. The keyassumption is that social networks, whose effects are captured in the parameter a, aremore tied to the ‘type’ of occupation than to the income of the worker.

The effort expended by young workers determines whether they do well in the lotteryto acquire White collar jobs; it is represented only in reduced form in our model. The factthat the probability of a Blue collar child’s accessing a White collar job is, ceteris paribus,less than the probability of a White collar child’s accessing a White collar job (that is,that a < 1) is the key circumstance that policy seeks to redress. Because this is acharacteristic of a type, it is appropriately called a circumstance rather than somethingdue to differential efforts of White and Blue collar children.

We do not claim to have established definitively that private financing of schoolsshould be abolished. First, we have not considered possible virtues of privately financedschools. Second, we have worked with a model that is merely an example. Nevertheless,due to the importance of equalizing opportunities for children through educationalpolicy, and the stark results of our model with respect to the possibility of equalizing

Economica


20 ECONOMICA [MAY

opportunities with different financing practices, we believe that the results should not beignored.

Finally, we are fully aware that a political analysis of how policy evolves, when not ina stationary state, would be very useful.4 Nevertheless, our analysis can be viewed asputting an upper bound on a political–economic analysis. With private investment ineducation, we have shown that in many cases at best, the stationary state may be laissez-faire: the stationary state to which a political–economic dynamic process converges (ifany) can only be worse. Without privately financed education, the history of actualNordic education and intergenerational mobility (for the latter, see Corak 2013) is,however, auspicious.

APPENDIX:

This appendix explains how we calibrate the threshold level for b given by World Indicators ofSkills for Employment, a dataset publicly available on the website of the OECD that provides thefrequency of high-skilled employees. In our notation, this datum corresponds to

a

1þ a:

The ratio of the average salary of highly skilled employees to the average wage is also providedby the same dataset. This datum corresponds to

ahW þ hWahW þ hB

:

We summarize in Table A1.

Routine calculations with these data yield the desired ratio 1/(ahW/hB + 1) as given in Table 5.In a similar fashion, these numbers imply

a ¼ 0:56 andhBhW

¼ 0:58

for the USA. Normalization by setting hW = 100 gives hB = 58. These give us the numbers used inthe simulation presented in the subsection of Section III entitled ‘Numerical analysis’.

TABLE A1DATA OF SELECTED COUNTRIES, SOURCE: WORLD INDICATORS OF SKILLS FOR

EMPLOYMENT, OECDWEBSITE

Sweden Norway Canada UK USA Germany France Japana

1þ a0.37 0.39 0.45 0.33 0.36 0.32 0.38 0.31

ahW þ hWahW þ hB

1.1 1.2 1.21 1.21 1.37 1.63 1.26 1.26

Economica



ACKNOWLEDGMENTS

We are grateful to an editor and two anonymous referees for their insightful comments andsuggestions that have led to significant improvements and changes in the paper relative to earlierversions. We also benefited from helpful discussions with seminar participants at Yale University,World Bank and Game Theory World Congress held at Istanbul Bilgi University.

NOTES

1. There is no political–economic element in this paper, which could determine how state policy is chosen ateach date. It is purely a welfare analysis.

2. Without this kind of structure, the number of income levels would become infinite over time, and the analysisof stationary states would become much more difficult.

3. One might conjecture that this view leaves the justly approved relationship between parent and child rathersterile. For a discussion of what one of us believes parents can pass on to their children, see Roemer (2012).

4. See Roemer (2006) for one such attempt.

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evidence. Economic Policy, 22, 781–861.CHETTY, R., HENDREN, N. and KATZ, L. (2015). The effects of exposure to better neighborhoods on children:

new evidence from the moving to opportunity experiment. NBERWorking Paper no. 21156.

CORAK, M. (2013). Income inequality, equality of opportunity, and intergenerational mobility. Journal of

Economic Perspectives, 27, 79–102.FLEURBAEY, M. (2008). Fairness, Responsibility andWelfare. Oxford: Oxford University Press.

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PERAGINE, V. (2004). Ranking income distributions according to equality of opportunity. Journal of Economic

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Documents

Dynamic Equality of Opportunity