Is 'tagging' a rationale for affirmative action in education?

FUNDAÇÃO GETULIO VARGAS

ESCOLA de PÓS-GRADUAÇÃO em ECONOMIA

Alexandre Balduino Sollaci

Is ‘Tagging’ a Rationale for Airmative

Action in Education?

Is ‘Tagging’ a Rationale for Airmative

Action in Education?

Dissertação para obtenção do grau de mes-tre apresentada à Escola de Pós-Grauação em Economia

Área de concentração: Taxação Ótima Orientador: Carlos Eugênio Ellery Lustosa da Costa

Ficha catalográfica elaborada pela Biblioteca Mario Henrique Simonsen/FGV

Sollaci, Alexandre Balduino

Is ‘tagging’ a rationale for affirmative action in education? / Alexandre Balduino Sollaci. - 2014.

55 f.

Dissertação (mestrado) - Fundação Getulio Vargas, Escola de Pós-Graduação em Economia.

Orientador: Carlos Eugênio Ellery Lustosa da Costa. Inclui bibliografia.

1. Programas de ação afirmativa na educação. 2. Impostos. I. Costa, Carlos Eugênio da. II. Fundação Getulio Vargas. Escola de Pós-Graduação em Economia. III. Título.

Agradecimentos

Neste artigo, procuramos racionalizar a existência de uma das formas mais comuns de polí-ticas de ação airmativa: cotas educacionais. Nós modelamos uma economia com dois períodos, assimetria de informação e formação endógena de capital humano, Os indivíduos dessa econo-mia podem vir de dois grupos diferentes, cada grupo deinido por uma característica exógena e observável. A distribuição de habilidades difere entre os dois grupos. Nós introduzimos cotas educacionais no modelo ao deixar o planejador social reduzir o custo, em termos de esforço, necessário para que um estudante de um desses grupos seja aceito numa universidade. Nesse contexto, uma política de ação airmativa pode ser interpretada como uma forma de tagging, já que as características de cada grupo podem ser usadas como proxies para produtividade. Concluímos que, embora políticas de cotas educacionais geralmente sejam eicientes, elas não necessariamente subsidiam a educação do grupo “menos"habilidoso.

Palavras-chave: ação airmativa, educação, tagging, taxação ótima.

Abstract

In this paper, we try to rationalize the existence of one of the most common airmative action policies: educational quotas. We model a two period economy with asymmetric information and endogenous human capital formation. Individuals may be from two diferent groups in the population, where each group is deined by an observable and exogenous characteristic. The distribution of skills difer across groups. We introduce educational quotas into the model by letting the planner reduce the efort cost that a student from one of the groups has to endure in order to be accepted into a university. Airmative action policies can be interpreted as a form of “tagging" since group characteristics are used as proxies for productivity. We ind that although educational quotas are usually eicient, they need not subsidize the education of the low skill group.

Contents

1 Introduction 1

2 Environment 5

3 The Benchmark Model 8

3.1 First-Best Solution . . . 8

3.2 Second-Best Solution . . . 11

4 Introducing Quotas into the Model 17 5 Possible Extensions 27 5.1 Inefficiency of quotas when there is discrimination on production . . . 28

6 Conclusion 30 A Proofs 34 A.1 The Benchmark Model . . . 34

A.1.1 First-Best Solution . . . 34

A.1.2 Second-Best Solution. . . 37

Introduction

Affirmative action is not a new subject, but it still engenders heated debates among its supporters and critics. As Fryer and Loury (2005) put, economic reasoning can help us shed some new light in the affirmative action debates and undo some commonly found misconceptions. Not surprisingly, there is a tantamount economic literature which concerns itself with affirmative action. The most seminal paper in this area is probably

Coate and Loury (1993), that shows how stereotypes may be self-fulfilling. Coate and Loury (1993) argue that if firms harbor a negative stereotype against workers from a given racial group, they will be less likely to assign these workers to higher paying jobs, which in turn undermines the incentives of workers from this racial group to invest in skill formation, and thus rationalizes firms’ beliefs. As in Arrow (1973), ex ante

discrimination is caused by a coordination failure in the labor market. Affirmative action in this case could serve to eliminate these (inefficient) negative stereotypes. However, quoting Coate and Loury(1993), “(...) there are equally plausible circumstances under which it [affirmative action] will not only fail to eliminate stereotypes, but may worsen them.” This latter case is what these authors call a patronizing equilibrium, where the favored group may undercut their investment in acquiring skills because they are favored in the first place.

Much of the subsequent literature is based on Coate and Loury’s insights. Moro and Norman (2003) extend their analysis to a general equilibrium framework, where wages are determined endogenously. They confirm the concerns present in Coate and Loury

(1993) and even prove that under some circumstances affirmative action may actually hurt its intended beneficiaries. Fang and Norman(2006) arrive at similar perverse results in a two sector general equilibrium model, driven by observations in Malaysia.

A second segment of the literature concerns itself with color-blind affirmative action. In this category, we find Chan and Eyster (2003), Fryer, Loury, and Yuret (2003) and

2

can proxy for race by using observable, nonracial characteristics that are not negatively correlated with a student’s performance. (...) At some point, though, this effort to find perfect proxies for race ceases to be “color-blind” in any meaningful sense.”

Empirical works in this subject generally take individual traits as given, and assess the effects of affirmative action on some specific group. Loury and Garman (1993),

Lott (2000), Bertrand, Hanna, and Mullainathan (2010), Francis and Tannuri-Pianto

(2012) and Ferman and Assun¸cao (2013) fit into this description. Finally, it is worth mentioning a detailed survey on theoretical advances in the affirmative action literature by Fang and Moro (2011) and a comprehensive text by Holzer and Neumark (2000) which covers historical and institutional background of affirmative action in the U.S. as well as a survey of the literature up to the time of its publication.

As evidenced above, most of the literature has focused on the effects of affirmative ac-tion on specific subgroups of the populaac-tion or in its role in eliminating stereotypes, while considerations on social welfare remain somewhat elusive. In fact, the existence of affir-mative action is sometimes only justifiable by some taste for racial diversity in schools or in the workplace. We try to shed some new light into the discussion by incorporating af-firmative action into the “New Dynamic Public Finance” literature (Kocherlakota,2010;

Golosov, Tsyvinski, and Werning,2007). We argue that an affirmative action policy is a form of tagging (Akerlof, 1978), and thus it relaxes the informational constraints to which the social planner is subject when designing a tax schedule. This is a particularly interesting exercise because we show, without having to assume any exogenous prefer-ence for diversity, that having affirmative action in education may be socially optimal. This result holds as long as the tagged and untagged groups are sufficiently different.1

The “New Dynamic Public Finance” literature has its roots on Mirrlees (1971), and is build upon the assumption that individuals have private information about their own productivity. Assuming the existence of a social planner whose objective is to maximize society’s welfare, this literature has made many important contributions to our understanding of how a tax system should work. We follow this tradition and further restrict ourselves to a two period economy with endogenous human capital formation, much like inKapicka and Neira (2014). Investment in human capital basically works in the following way. In the first period, individuals choose if they should invest in human capital (study) or not, at the cost of effort. We assume that individuals who invest in human capital are the ones who attend college or university. In order to be accepted into and to graduate from a college or university, individuals have to study past a fixed effort threshold (for example, they may have to pass an admission exam and the mandatory courses for graduation). For simplicity, we take this threshold as uniform and given. If a student is accepted into and graduates from college, he becomes more productive in the second period.

To introduce affirmative action into the problem, we focus on affirmative action in higher education. The instrument available to the planner is establishing quotas. We abstract from efficiency issues concerning different affirmative action instruments2

1

This difference will be made clear in chapter 3.

2

by arguing, asFryer and Loury (2005), that “any effective enforcement policy will have quota-like effects”. Educational quotas generally establish that a minimum percentage of students admitted into universities are from a given group. One of the practical effects of this policy is that a benefited student may be admitted into a university without having to exert as much study effort as another identical student from a different group. Since the number of students per class remains unchanged, a quota will, in the terms of our model, lower the effort threshold for being accepted into a university for the targeted group and at the same time raise the same threshold for the other group. But this is the same as taxing the effort of one group and subsidizing the effort of another! Hence, implementing an educational quota is no different than tagging one group and “discriminating” taxes on effort between groups.3

The idea behind tagging, first present in Akerlof(1978), is that, if there is an exoge-nous characteristic (such as skin color) that is correlated with individual productivity, then the planner should use this characteristic to identify (tag) individuals and infer information about them. This will relax the informational constraints to which the planner is subject and, if used in an optimal way, tagging increases social welfare. How-ever, tagging is almost always thought of as discriminating the income tax schedule between groups. In this case, albeit the large potential gains from tagging (see Cremer, Gahvari, and Lozachmeur, 2010), it is seldom observed in practice. Weinzierl (2012) suggests that a reason for this is that an utilitarian social function does not capture the whole spectrum of what society considers as “justice”. In particular, tagging violates principles such as horizontal equality and equal sacrifice (which basically call for equally treating equals), and therefore it faces moral barriers for its implementation.

Nonetheless, if our interpretation of an educational quota as discriminating taxation is not too unrealistic, then we may argue that tagging does exist in practice, in the form of affirmative action in higher education. A possible justification for the better popular acceptance of tax discrimination on effort instead of on income is that, following the argument of Rubenfeld (1997), discriminating taxes on effort denies the non-targeted group potential opportunities to which individuals are merely applying, while discrim-inating taxes on income takes from them what they already have. Justice issues aside, what we highlight in this essay is something that people have repeatedly failed to realize in the debate on affirmative action: quotas are an analogue to tagging a specific group, and may be an efficient way to use the information carried by this tag.

A second issue that we address in this paper is whether educational quotas would remain optimal if the planner were able to discriminate income and consumption between groups. In other words, we ask if affirmative action just a socially acceptable proxy for discriminating income taxes between groups. Using a very simple argument we find that this is the case. Whenever possible, a social planner will always prefer to discriminate consumption and production allocations instead of discriminating effort between groups.

vouchers. He advocates that targeted vouchers lead to higher quality of the work force, more efficient allocation of resources and greater social mobility.

3

4

There is a surprisingly small number of papers that deal with affirmative action from an optimal taxation point of view, despite the importance of both these literatures.

Blumkin, Margalioth, and Sadka(2009) is probably the closer to this paper, discussing the redistributive role of affirmative action in a Mirrleesian framework. However, this paper is different from theirs in several aspects. First of all, they analyze affirmative action in the labor market, and we focus affirmative action in education, in the form of quotas. Second, our framework allows for a continuum of individuals, while theirs has only two levels of productivity. Finally, they have a clear distinction between affirmative action (which they model as an additional constraint that there is racial equality among the employees of a firm) and differentially taxing individuals from different groups (tag-ging). In our framework, affirmative action and tagging confound themselves, since we model an educational quota as a tax/subsidy on study effort.

Environment

We model a two period economy with a continuum of individuals (whose measure we normalize to unity) who have private information about their own productivity level. Individual productivity is given by the one-dimensional parameter θ _∈ [θ,θ] (where

θ > 0 and θ < _∞) and distributed according to the distribution function F(θ) with densityf(θ). The population in this economy can be divided into two groups, group A

and group B, that can be identified by a exogenous characteristic (e.g. racial heritage or sex). Individual productivity in group A and in group B is distributed according to, respectively, fA(θ) andfB(θ). We also assume that group A takes a shareπ of the

population and groupB takes a share (1₋π), so that f(θ) =πfA(θ) + (1−π)fB(θ).

In the first period, individuals may invest in human capital.1 In order to do so, an individual must attend college or university. If he successfully graduates from college, he becomes more productive in the following period: his productivity increases from θ to

hθ, where h >1. To graduate from college, a minimum level of efforteis required. This is what we call productive education effort, since it translates into higher productivity in the future.

However, the representative university2 in this economy cannot admit everyone for lack of physical capacity (think of this whole analysis as a ‘short term’ exercise, where there is no room for enlarging the university, for example). We introduce this restriction by allowing no more than a measure µ of students to be enrolled at any given time. If this constraint binds, the university adopts a selection process that is costly to individu-als. For example, the university may require that candidates take an admittance exam, and only those with the highest scores are admitted. We will assume that, regardless of the selection process, the university will accept individuals based on an effort thresh-old: whoever is above this threshold is accepted into college and whoever is below the threshold is not.

1

The fact that individuals may only invest in human capital in the first period is without loss of generality. Nobody would invest in human capital in the second period because there are no gains from it.

2

6

An important point here is that the planner may want to assign a different threshold for individuals from different groups. For example, individuals from group A may be admitted if they are above the effort threshold εA, while individuals from group B

are admitted if they are above the threshold εB. Note that producing an effort equal

to e is already enough for an individual to become more productive; hence, εi, i ∈

{A,B_}, is socially wasteful effort that only serves the purpose of screening individuals (a signal). This assumption will be better discussed later on, but it is particularly important because it allows us to have quotas without compromising the human capital gains from education.

We aggregate the effort needed for being accepted to college and that for completing the mandatory courses in one single effort threshold in each group, ei =εi+e, which

we call the absolute cost of investing in human capital. In other words, producing a level of effort equal to or greater than ei is enough for one individual from group i to

get into college and to graduate from it. Also, given that there is heterogeneity in the population, some individuals may find it easier to exert effort than others. Assuming that more productive individuals are the ones who find studying easier, therelative cost (or utility cost) of investing in human capital isei/θ, fori∈ {A,B}.

Furthermore, effort itself is not observable by the planner, although human capital “status” is. This means that the planner cannot directly designate an effort allocation to individuals, but he does know if an individual chose to invest in human capital by attending college. Hence, while investing in human capital is an individual choice, once this decision is made, the planner knows if an individual has chosen to exert effort or not, and can thus condition other allocations on this decision.

In this economy, there exists only one consumption good that can be produced using skill weighted labor and a one-to-one production function. Individuals care for consump-tion and dislike labor. In each period t, we denote the consumption and labor supply of an individual with productivity θ by ct(θ) and ℓt(θ), respectively. ℓt is not publicly

observable and ℓt∈[0,ℓ], t∈ {1,2} – that is, individuals have a limited amount of time

available for them to work. As discussed before, the planner cannot discriminate be-tween groups when assigning consumption-production bundles to individuals because of political issues. However, this is not true for the labor supply. In fact, If the planner wishes to assign different effort thresholds to people from different groups, he will also have to assign different labor supplies for individuals with the same productivity but from different groups. When this is the case, we will represent the labor supply of an individual with productivity θfrom groupiand in period t byℓi_t(θ).

Letyt(θ) denote what an individual with productivityθproduces of the consumption

good with labor supply ℓt(θ) in period t. As described above, the consumption good

can be produced using skill weighted labor and a one-to-one production function. Hence

θℓt(θ) =yt(θ), fort= 1,2, if individualθdoes not invest in human capital. If, however,

individualθstudies in the first period, we haveθℓi₁(θ) =y1(θ)+ei,i∈ {A,B}(individuals

have to sacrifice production in order to study), and hθℓ2(θ) = y2(θ) (but, in exchange,

W(θ) =U(c1(θ))−

y1(θ)

θ +βU(c2(θ))−β y2(θ)

θ

if the individual does not study in the first period, and by

W(θ,e) =U(ce₁(θ))₋y

e

1(θ) +e

θ +βU(c

e

2(θ))−β

y₂e(θ)

hθ

if he does.

As above, we will use the superscript e to indicate that the variable in question is allocated to an individual who invests in human capital. For example, ce

t(θ) indicates

the consumption of an individual with productivityθthat has invested in human capital and ct(θ) indicates the consumption of an individual with productivity θ who has not

invested.

Finally, we also assume that U is continuously differentiable, strictly increasing and strictly concave. These assumptions will guarantee that, if there a solution to the plan-ner’s problem, the first-order conditions will be sufficient conditions to find them. To guarantee that there is a solution to the planner’s problem, we will assume that

U(0)> U(c(θ))₋ℓ

for any feasible c(θ) _≥ 0. This assumption is the analogous to assumption [A-3] in

Berliant and Page(2001) that states that leisure is essential. Basically, this hypothesis states that, to every individual, it is preferable not to work in exchange for no con-sumption than to work for their entire time endowment in exchange for any feasible consumption level (by feasible consumption we mean a level of consumption that can be produced by the individual).

Chapter 3

The Benchmark Model

In this chapter, we will present and solve a benchmark model in which there is no possibility of implementing an affirmative action policy. In this case, the planner cannot discriminate between groups in any way – in particular, we have εA=εB≡ε.

3.1

First-Best Solution

It is useful to characterize the optimal allocations when the planner observes individual productivity and effort. We will solve this problem in two steps. First, we find the optimal allocations for every individual in this economy when he studies and when he doesn’t. Second, we find out, for each individual, if the planner recommends that he exerts effort or not.

Letχindicate if individuals study or not in the social optimum; that is,χis a dummy variable representing the effort decision of each individual:

χ=

1 if the individual studies; 0 otherwise.

By choosing the appropriate value for α, we can write h = 1 +α, so that the first best problem can expressed as

max

{ct(θ),yt(θ)}t=1,2 Z θ

θ

U(c1(θ))−

y1(θ) +χe

θ +β

U(c2(θ))−

y2(θ)

θ(1 +χα)

f(θ)dθ

s.t.

Z θ θ {

y1(θ) +βy2(θ)−c1(θ)−βc2(θ)}f(θ)dθ≥0.

investment in human capital, given his productivity. By doing that, we can determine the optimal allocation of effort. Only then we incorporate the capacity constraint to see how it changes the optimal allocation of effort.

Proposition3.1 describes the optimal allocations.

Proposition 3.1. In the first-best optimum: (1) Consumption is constant across individuals,

c= (U′)−1(λ),

whereλis the multiplier on the resource constraint.

(2) Production is given by

y1(θ) =

θℓ₋χe, ifθ_{≥ λ}1; 0, otherwise.

y2(θ) =

(

θℓ(1 +χα), ifθ_{≥ λ(1+}1_χα); 0, otherwise.

(3) The optimal effort allocation is given by one of the following: If the capacity constraint is not binding and

• λe _≤ βℓ(h₋1), the planner will allocate effort based on a cutoff rule. If the capacity constraint is not binding, everyone with productivity above θ0 should

invest in human capital, where

θ0 =

βℓ+pβ2_ℓ¯2_{+ 4βℓhλe}

2βℓhλ .

• λe > βℓ(h₋1), then it is optimal that individuals with productivity in two disjoint intervals invest in human capital. If the capacity constraint is not binding, it is optimal that individuals with productivity in [θ0,1_λ]∪[_βℓ(he₋₁₎,θ]

invest in human capital. Figure 3.1illustrates this case.

If the capacity constraint is binding and

• the planner does not follow a cutoff rule when assigning effort, the best that he can do is to recommend that the least productive people in each of the tow disjoint intervals do not study; but when comparing individuals with produc-tivities in different intervals, he should follow the ratio: for every individual with productivity θ _∈ [θ0,1_λ) that does not study, _βℓhλβℓ(h₊−_e/θ1)2 individuals from

the group [ e

10

• the planner follows a cutoff rule when assigning effort, he will first assign no effort for the least productive individuals with productivity in [θ0,_λ1) until he

reaches the point θ1, where θ1 is the point that equals the social surplus of

an individual with productivity in [θ0,_λ1) with the social surplus generated by

and individuals with productivity equal to _λ1. If the capacity constraint is met before then, this will the optimal effort allocation. If not, he will continue as in the last case: for every individual with productivity θ _∈ [θ1,_λ1) that does

not study, βℓ(h−1)

βℓhλ+e/θ2 individuals from the group [

1

λ,θ] also should not invest in

human capital.

Proof. See AppendixA.1.1

θ

Social Surplus

−e

• 1λ

1

hλ

e βℓ(h−1)

θ2βℓhλ₋βℓθ₋e

βℓ+√β2_ℓ¯2_+4βℓhλe

2βℓhλ

Figure 3.1: One possible effort allocation throughout the population: in red are individ-uals who do not invest in human capital and in blue are the individindivid-uals who invest. We ignore negative values of productivity.

Note that when it is optimal not to have a cutoff rule and the capacity constraint is not binding, the utility difference between someone who does not invest in human capital but works in both periods —θ_∈(1/λ, e/βℓ(h₋1)) — and someone who invests in human capital but does not work in the first period – ˆθ_∈(θ0,1/λ) – is −_θℓ −e_θˆ. For

effort is not observed by the planner); in this case, the effort allocation rule throughout the population will always be a cutoff rule.

3.2

Second-Best Solution

We now focus our attention in solving the planner’s problem when individuals have pri-vate information about their own productivity and effort decision. As argued before, the planner can still observe if an individual has exerted effort above or below the threshold

e. This does not mean that the planner directly observes effort, but rather that he has some instrument that screens individuals based on this effort threshold, such as being en-rolled in college or not. Allocations in this setting are a sequence_{(ct(θ),yt(θ))2t=1}θ∈[θ,θ]

with consumption and production in both periods being a function of an individual’s productivity and perhaps of his effort choice (above or below the effort threshold e).

The key feature of a setting where productivity is not observable by the planner is that allocations have to be incentive compatible; that is, an individual must find the allocation designed for him preferable to any other allocation, given his own productivity. We can use the Revelation Principle to restrict ourselves to direct mechanisms without loss of generality (see Myerson (1982)). Notice, however, that the planner has two informational issues to address. Because allocations of consumption and production assigned by the planner to each individual may depend on whether that individual has chosen to study or not, the planner has to make sure that individuals do not have incentives to misrepresent their true productivity in two ways. The first one is the traditional way, in which, given an effort allocation, an individual must maximize his utility by correctly revealing his own productivity. The second one is particular to this setting, and captures the idea that allocations have to be inventive-compatible even between individuals who choose different effort allocations. Take an individual with productivity θ who chooses not to study. Then, he must find it optimal not to mimic someone with productivity ˆθ when this other person chooses not to study and when this person chooses to study. This could imply that we need to be concerned with global properties of the incentive compatibility constraint. Fortunately this is not the case.1 To write down the incentive compatibility constraint, we will first introduce some notation. Let W(ˆθ_|θ) represent the lifetime utility of an individual with productivity

θ but who announces a productivity ˆθ and does not invest in human capital; that is

W(ˆθ_|θ) = P2_t=1βt−1_[U(c

t(ˆθ))− yt_θ(ˆθ)]. Analogously, W(ˆθ,e|θ) = P2_t=1βt−1U(cet(ˆθ))− ye

1(ˆθ)+e

θ −β ye

2(ˆθ)

hθ (Note that W(θ|θ)≡W(θ) and W(θ,e|θ)≡W(θ,e)). Finally, define by

V(ˆθ_|θ) the maximum utility an individual with productivityθcan achieve by announcing that his true productivity is ˆθ: V(ˆθ_|θ) = max_{W(ˆθ_|θ), W(ˆθ,e_|θ)_}. In this case, the incentive compatibility constraint is:

1

In our setting, the planner cannot directly assign effort allocations to individuals, which simplifies the problem. For a discussion on the effects of simultaneous adverse selection and moral hazard, see

12

V(θ_|θ) = max

max

ˆ

θ∈[θ,θ]

W(ˆθ_|θ), max

ˆ

θ∈[θ,θ]

W(ˆθ,e_|θ)

, _∀θ.

In order to solve the second-best problem, we will again adopt a two step strategy. First, we will present sufficient conditions that guarantee that the curves representing individual utility when studying and when not studying only cross exactly once, so that individuals choose their effort allocation based on how their productivity compares with the point in which both these curves cross. This allows us to write the second-best problem almost as two separate problems, which is more convenient to find the optimal allocations. We then follow Kapicka and Neira (2014)’s strategy in solving the second-best problem by the first-order approach.

First, we use the following Lemma to find envelope constraints (B-EC1) and (B-EC2).

Lemma 3.1. Let y1(θ), y2(θ), y1e(θ) and ye2(θ) be non-decreasing functions of θ for all

θ_∈[θ,θ] andW(θ) andW(θ,e) be differentiable functions ofθ for almost all2 _{θ∈(θ,θ).}

Suppose further thatW(θ) andW(θ,e) are right-hand differentiable atθ and thatW(θ

and W(θ,e) are left-hand differentiable at θ. Then

W(θ) = max

ˆ

θ∈[θ,θ]

W(ˆθ_|θ)_⇔W(θ) =W(θ) +

Z θ θ

"

y1(˜θ)

˜

θ +β

y2(˜θ)

˜

θ

#

dθ˜

˜

θ (B-EC1)

and

W(θ,e) = max

ˆ

θ∈[θ,θ]

W(ˆθ,e_|θ)_⇔W(θ,e) =W(θ,e) +

Z θ θ

"

ye

1(˜θ) +e

˜

θ +β

ye

2(˜θ)

hθ˜

#

dθ˜

˜

θ .

(B-EC2)

Proof. See AppendixA.1.2

This envelope form of the incentive compatibility constraints is useful for two rea-sons. First, it allows us to solve the second-best problem through a simple pointwise maximization of a Lagrange function. Second, it explicitly shows how the utilityW of an individual responds to variations in his productivity and effort decision. In particular, we use the envelope form of the IC’s to prove the next proposition, which will help us show that individuals choose to invest in human capital by comparing themselves to a cutoff point.

Proposition 3.2. Suppose that all the conditions listed in Lemma 3.1 hold, and let

y1(θ),y2(θ),y1e(θ) andy2e(θ) also be continuous functions ofθ. Suppose further that there

exists someθsuch that Wθ(θ)< Wθ(θ,e). Then, for all ˆθ > θ, we haveWθ(ˆθ)< Wθ(ˆθ,e)

in every incentive compatible allocation.

2

Proof. See AppendixA.1.2

If W(θ) > W(θ,e),3 Proposition 3.2 basically states that in the case that W(θ) and W(θ,e) “cross”, then they only cross once. To guarantee that these functions do cross, we will further assume that W(θ,e) > W(θ). This is of course with some loss of generality, but in the event that these two functions never cross, we would wind up with an uninteresting case and would, in particular, have no room for discussing affirmative action policies.

Because Proposition 3.2 is a consequence of the incentive compatibility constraints, we will have a unique point in whichW(θ) andW(θ,e) cross in every incentive compatible allocation (assuming thatW(θ)> W(θ,e) andW(θ,e)> W(θ) hold). In particular, this will happen with the socially optimal allocation of consumption and production. Hence, in a social optimum, we will have amarginal individual, with productivityθm, such that

W(θm) = W(θm,e). Individuals whose productivity is θ ≥ θm will choose to invest in

human capital, while individuals whose productivity is θ < θm will choose not to.

Finally, note that the optimal allocations may depend on who invests in human cap-ital in equilibrium. Intuitively, a person who invests in human capcap-ital becomes more productive in the second period and therefore produces more with the same labor out-put. This makes the economy’s resource constraint – and therefore the allocations of consumption and production – depend of the allocation of effort. On the other hand, individuals’ choice of investment in human capital depends of their consumption and production allocations. As a result, although the cutoff rule remains optimal in ev-ery incentive-compatible allocation, either the cutoff point or the effort threshold (or productivity boost, h) are endogenous to the problem.

Now that we have established the validity of the cutoff rule, we define two separate problems (connected by the resource constraint of the economy): one for individuals who invest in human capital, and one for individuals who do not. First of all, the incentive compatibility constraints can be written as

W(θ) =W(θ) +

Z θ θ

"

y1(˜θ)

˜

θ +β

y2(˜θ)

˜

θ

#

dθ˜

˜

θ , θ∈[θ,θm); (B-EC1’)

W(θ,e) =W(θm,e) +

Z θ θm

"

ye₁(˜θ) +e

˜

θ +β

y₂e(˜θ)

hθ˜

#

dθ˜

˜

θ , θ∈[θm,θ] (B-EC2’)

since we know that individuals with productivity belowθm will maximize their utility if

they do not study and individuals with productivity aboveθmwill maximize their utility

if they do.

However, we also need to require that

W(θm) =W(θm,e). (B-EC3)

3

14

The expression in (B-EC3)4 guarantees incentive compatibility at the cutoff point, θm.

In fact, if (B-EC3) does not hold, then the utility of individuals is not continuous in

θm, which is not incentive compatible.5 In addition, constraint (B-EC3) captures the

discussion in the preceding paragraph by defining either e or θm as an endogenous

variable in the problem and making the relation between these two variables explicit. Furthermore, note that we can merge (B-EC2’) and (B-EC3) into the single expres-sion

W(θ,e) =W(θm) +

Z θ θm

"

ye

1(˜θ) +e

˜

θ +β

ye

2(˜θ)

hθ˜

#

dθ˜

˜

θ , θ∈[θm,θ). (B-EC2”)

Also, since we know that individuals on this economy will choose whether or not to invest in human capital by using a cutoff rule, the restriction of physical capacity in universities can be written as

1₋F(θm)≤µ (B-CC)

– that is, the measure of individuals enrolled must be smaller or equal toµ, the physical capacity of the university. In everything that follows, we will suppose that this constraint binds at the optimum.6 It makes no sense to speak about educational quotas (as we will in the next chapter) when universities are not at full capacity. If this was the case, instead of having a quota for some group, we could just increase the number of enrolled students from a targeted group and no trade-off would be at work.

If restriction (B-CC) is binding, then the social planner has to induce exactly 1₋

F(θm) individuals to choose to study. The first, and perhaps more immediate way to do

this, is to choose an appropriate value for the effort threshold. For example, suppose that the minimum study effort necessary to increase an individual’s productivity by a factor ofhise. Suppose also thateis small enough so that (given consumption and production allocations) a fraction of the population greater than µ chooses to study. In this case, the planner could simply increase the effort threshold to e > e by adding a signaling component to the effort threshold. This would still be incentive compatible and would discourage the least productive people from studying. By choosing the appropriate value fore, we could induce exactly µpeople to study.

However, because adding a signaling component to the effort threshold is an unpro-ductive cost, this is inefficient. Suppose that individuals who study produce y₁e(θ) in the first period and the effort threshold is e (including the signal). The planner could then ask that individuals produce instead ˆy₁e(θ) =ye₁(θ) +e₋e. This is also incentive

4

Given the cutoff rule,W(θm) won’t be well defined in the social optimum because we’ve assumed

that individuals with productivity greater or equal to θm will invest in human capital. Hence, define

W(θm) = limθ→θmW(θ) whenever necessary. Keep this in mind for similar occurrences throughout the

paper.

5

Note that the utilityW must be continuous at all points, despite the fact that neither the actual consumption nor production are. In fact, we expect consumption and production to be discontinuous at the point where individuals change their human capital investment, exactly so that individual utility is continuous at this point.

6

compatible, induces exactlyµindividuals to study, and generates a surplus which can be then redistributed, increasing everyone’s utility. In the problem without quotas, this is an efficient mechanism available to the planner to comply with (B-CC). In the problem

with quotas, where the planner may want to induce a different cutoff for different groups of the population, this is no longer possible if he is restricted not to “discriminate” taxes between groups. In this case, the sole instrument available to induce individuals to invest (or not) in human capital is the addition of an appropriate signal to the effort threshold.

Having established all that, the second best problem can now be written as

max

{ct(θ),yt(θ),cet(θ),yet(θ)}t=1,2 Z θm

θ

W(θ)f(θ)dθ+

Z θ θm

W(θ,e)f(θ)dθ

s.t.

Z θ θ

[y1(θ) +βy2(θ)−c1(θ)−βc2(θ)]f(θ)dθ≥0 (B-RC)

W(θ) =W(θ) +

Z θ θ

"

y1(˜θ)

˜

θ +β

y2(˜θ)

˜

θ

#

dθ˜

˜

θ , θ∈(θ,θm) (B-EC1’)

W(θ,e) =W(θm) +

Z θ θm

"

y₁e(˜θ) +e

˜

θ +β

y₂e(˜θ)

hθ˜

#

dθ˜

˜

θ , θ∈[θm,θ) (B-EC2”)

1₋F(θm) =µ (B-CC)

Let λbe multiplier on the resource constraint (B-RC)7, δ(θ)f(θ) the multiplier on (B-EC1’),γ(θ)f(θ) the multiplier on (B-EC2”) andφthe multiplier on (B-CC).

To find the optimal second-best allocations, we take the first-order conditions of the Lagrangian in Lemma A.1 with respect to individual consumption to find the optimal consumption levels. We then choose the production levels to solve the equations given by envelope form of the incentive compatibility constraints. Finally, by taking first order conditions with respect to individual production, we show that there might be bunching in the lower levels of productivity.

Proposition 3.3. Assume that the conditions listed in Lemma 3.1are satisfied. Then, in the solution of the second-best problem:

(1) The Euler equation holds:

U′(c1(θ)) =U′(c2(θ)) = ₁₊λ_δ(θ) ∀θ < θm.

U′_(ce

1(θ)) =U′(ce2(θ)) = 1+λγ(θ) ∀θ≥θm.

7

We have abused notation in the representation of the resource constraint of the economy. Ideally, it should be written as Rθm

θ [y1(θ) +βy2(θ)−c1(θ)−βc2(θ)]f(θ)dθ+

Rθ θm[y

e

1(θ) +βye2(θ)−ce1(θ)− βce2(θ)]f(θ)dθ≥0. However, in the interest of visual presentation, we will keep representing the resource

16

(2) One optimal production allocation will be given by:

y1(θ) =

1 2

(1 +β)θU(c(θ))₋θW(θ)₋(1 +β)

Z θ θ

U(c(˜θ))dθ˜

y2(θ) =

1 2β

(1 +β)θU(c(θ))₋θW(θ)₋(1 +β)

Z θ θ

U(c(˜θ))dθ˜

.

y₂e(θ) = min

ℓθh,h β

(1 +β)θU(ce(θ))₋θmW(θm)−(1 +β)

Z θ θm

U(ce(˜θ))dθ˜₋e

and

y₁e(θ) = max

0,(1 +β)θU(ce(θ))₋θmW(θm)−(1 +β)

Z θ θm

U(ce(˜θ))dθ˜₋e₋βy

e

2(θ)

h

,

where U(c(θ)) = U(c1(θ)) = U(c2(θ)), U(ce(θ)) = U(ce1(θ)) = U(ce2(θ)).

Further-more, if there is bunching (see below), we have

W(θ) = (1 +β)U(λ).

(3) There is bunching in the lower productivity levels if θ < 1 +δ(θ) + ∆(θ)

λ or if θ <

1 +γ(θ) + Γ(θ)

hλ .

Proof. See AppendixA.1.2

Finally, even though the cutoff point is already determined by restriction (B-CC) as

θm=F−1(1−µ), it is useful to analyze the first-order condition with respect to it. When

we implement an educational quotas policy, we are in fact choosing two different cutoff points for people from different groups in the population; hence, the first-order condition with respect toθm in the problem without quotas may help us gain some insight into the

tradeoffs that we face when varying the cutoff point. By setting the FOC with respect to

θm equal to zero, we find (see the proof of Proposition4.1 for an analogous derivation):

[Wθ(θm,e)−Wθ(θm)]

Z θ θm

γ(θ)f(θ)dθ =f(θm){λ[Se(θm)−S(θm)]−δ(θm)[W(θm)−W(θ)]−φ}

or

Z θ θm

γ(θ)f(θ)dθ= f(θm){λ[S

e_(θ

m)−S(θm)]−δ(θm)[W(θm)−W(θ)]−φ}

Wθ(θm,e)−Wθ(θm)

.

This equation tells us that there are two distinct effects of a variation in the cutoff point. The first effect is local, and given by the right-hand side of the equation above. It is, in general terms, the social cost of letting f(θm) additional people into college.

Introducing Quotas into the

Model

To introduce the possibility of educational quotas into our model, the first thing to do is to allow for some observable diversity in the population. Suppose that the population is divided into two groups, A and B, characterized by an exogenous and observable characteristic. These two groups could be differentiated by racial heritage, gender or even height – as long as the characteristic that defines each group is (mostly) exogenous. Suppose further that group A takes a share π of the population, and group B takes a share 1₋π.

We will also assume that the distribution of productivity in each group is different. Productivity is distributed in group A according to the marginal distribution function

fA(θ) and in group B according to fB(θ). Hence, f(θ) = πfA(θ) + (1−π)fB(θ). This

is a delicate point, and needs to be interpreted correctly. We do not mean to imply that people from different population subgroups are naturally more or less capable than others. Instead, a different productivity distribution between groups is just a convenient way to model group discrepancies observed in practice. Take, for example, the persistent black-white test score gap (seeCard and Rothstein, 2007). It is one undeniable evidence of how white and black people, on average, differ in educational achievement. Heckman and Cunha (2007) find evidence of a series of factors that, especially during an early age, largely impact a person’s ability. These factors include parental investment and the child’s environment, which tend to vary a lot between white and black families, due to socioeconomic factors, racial discrimination or even some negative self-image as a group. Our hypothesis of different productivity distributions between groups is an attempt to capture these underlying differences between groups in the simplest way possible. In fact, as long as these groups’ differences are correlated with educational achievement and observed production, they can be modeled through a productivity parameter.

18

to be accepted into a university is merely a signal and does not affect productivity in the second period. Hence, we interpret a quota policy as a program that introduces different signaling costs between the two groups,εAandεB. Once again, this is possibly

a controversial point. However, it finds resonance in empirical works. Let us break the argument into two parts.

The first part of the argument is that individuals who enter universities through a quotas system need to exert less effort than those who enter through regular ways. This is intuitive, as long as we keep in mind that we are referring to the effort threshold and not to what we called the relative effort (the utility cost of exerting effort). This distinction is important, and may even reconcile apparently contradicting results in the empirical literature. Francis and Tannuri-Pianto (2012), for example, find evidence that the racial quotas did not reduce the pre-university effort of (benefited) applicants to the University of Brasilia (in Brazil). Ferman and Assun¸cao (2013), on the other hand, find evidence of the contrary in universities of the States of Rio de Janeiro and Bahia (also in Brazil). The difference between these two results may come from the difference between relative and absolute effort: Francis and Tannuri-Pianto(2012) measure effort (in general terms) as the time spent studying for admittance exams, which is a measure of relative effort;Ferman and Assun¸cao (2013) use test scores as a proxy for effort, and thus measure how a student compares with others – or, in other words, how a student compares with the expected effort threshold in admittance exams.

We cannot say much about the effects of quotas on relative effort in a given group,1 but it is probably not unrealistic to assume that the standards for admission are lowered in the benefited group (and therefore raised in the other group) – in fact, this is the whole purpose of the quota policy.

The second part of the argument is that part of the effort required to be accepted into college does not make individuals more productive – that is, it is merely a signal. There is plenty of evidence that the scores in admittance exams to universities are not good predictors of a student’s later academic achievements. Rothstein (2004) shows that this is the case in the USA, where students must take the SAT. Pereira (2013) and Velloso (2009) both find evidence that Brazilian students who entered university through the quotas system did not have a systematically worse academic achievement (at university) than students who entered university without the quotas system, despite having systematically lower scores in the admittance exams. There is, of course, still some difference between white and black student’s academic achievement at university. However, it can be easily attributable to different skill distributions between whites and blacks or even to different choices that white and black students make, on average, at university (for example, Francis and Tannuri-Pianto (2012) find evidence that black students tend to apply to ‘less selective’ courses than white students do).

In a nutshell, our point is that if the effort needed to pass the admittance exam did in

1

fact make individuals more productive, we would expect students with the highest scores in these exams – white students – to have, on average, greater academic achievements than students with low scores – black students. This is clearly not the case, and therefore the empirical evidence indicates that at least part of the effort exerted for admittance works just as a signal of the individual’s productivity, rather than as productive effort.

The final ingredient of the problem with the possibility of educational quotas is the introduction of a restriction of physical capacity of universities. That is, a social planner cannot simply let more people from a given group into college due to lack of physical capacity in universities – at least in the short term. Hence, if he is to facilitate the admittance of people from a given group into college, he must compensate by making it harder for people from the other group to be admitted. In mathematical terms, this would mean that the measure of individuals who invest in human capital does not change after the introduction of affirmative action into the problem, so that we have the following capacity constraint

F(θm) =πFA(θAm) + (1−π)FB(θmB). (Q-CC)

In words, the fraction of the population that did not study in the problem without quotas (F(θm)) is equal to the fraction of the population who does not study in the problem

with quotas, but the group composition of this fraction may change.

All that we have done up to this point is to introduce the possibility that the planner asks universities to accept students from a given group with a lower signal than they would without an affirmative action policy. Hence, most of the derivations we made in the problem without quotas remain true in the problem with quotas, although they are valid only inside each group. In particular, let y1(θ), y2(θ), y1e(θ) and y2e(θ) be

non-decreasing functions and continuous of θ for allθ_∈[θ,θ] andW(θ) and W(θ,e) be differentiable functions ofθfor almost allθ_∈(θ,θ) and alle. Suppose further thatW(θ) and W(θ,e) are right-hand differentiable at θ and all e, and that W(θ and W(θ,e) are left-hand differentiable atθand alle. Then the cutoff rule still holds in each group, and the incentive compatibility constraints can be written as

W(θ) =W(θ) +

Z θ θ

"

y1(˜θ)

˜

θ +β

y2(˜θ)

˜

θ

#

dθ˜

˜

θ , θ∈[θ,θ

i

m) (Q-EC1-i)

W(θ,ei) =W(θim,ei) +

Z θ θi

m "

ye₁(˜θ) +ei

˜

θ +β

y₂e(˜θ)

hθ˜

#

dθ˜

˜

θ , θ∈[θ

i

m,θ] (Q-EC2-i)

and

W(θi_m,ei) =W(θim) (Q-EC3-i)

fori_{∈ {}A,B_}. Following the notation of the last chapter,θi_m is the productivity of the individual in groupiwho is indifferent between studying or not.

20

Corollary 4.1. Suppose that the planner cannot discriminate between groups (that is, allocate consumption-production bundles to individuals with the same productivity but from different groups). If the planner wishes to introduce an educational quota, he must distort the margin of who studies and who does not study between groups. Conversely, if the planner wishes to distort the margin of who studies and who doesn’t, he has to introduce an educational quota (reduce the effort threshold for one of the groups).

Proof. (Corollary 4.1) This is a direct consequence of the requirement that W(θ_mi ) =

W(θi_m,ei), which guarantees that the utility of individuals is continuous onθim.

(eA6=eB⇒θAm6=θmB):

To prove the first part of the corollary, suppose that eA 6= eB, but θAm = θmB = θm.

ThenW(θm,eA)=6 W(θm,eB). But we must haveW(θm,eA) =W(θm) andW(θm,eB) =

W(θm). Hence,W(θm,eA) =W(θm,eB) which is a contradiction. Therefore,eA6=eB ⇒

θA_m6=θ_mB. (θA

m6=θmB ⇒eA6=eB):

To prove the converse, suppose eA=eB =e. Then, for a given productivity, the utility

of an individual from group A is the same as the utility of an individual from group B

(because the planner cannot discriminate consumption and production bundles between groups). Because individual utility is strictly increasing in productivity,2 _{if θ}A

m is the

cutoff point for groupAand θB

m is the cutoff point for groupB, we must haveθAm =θmB.

In fact, if this was not the case, then we would haveW(θ_mA) = W(θ_mA,e) andW(θB_m) =

W(θB

m,e) with θmA 6= θBm. From Proposition 3.2, this is impossible, because W(θ) and

W(θ,e) would have to cross in two distinct points.3 _Thereforee

A=eB ⇒θmA =θBm, and

we conclude that θ_mA ₆=θ_mB _⇒eA6=eB.

Note that Corollary4.1implicitly defineseAas a function ofθAmandeBas a function

ofθB_m (or vice versa, by inverting these functions). This is basically the same restriction in the cutoff point that we had in the second-best problem without the possibility of tagging, but now we must abide by it in both groups.

Once again, following the procedure of the problem without quotas, we can condense (Q-EC2-i) and (Q-EC3-i) into the single expression

W(θ,ei) =W(θim) +

Z θ θi

m "

ye₁(˜θ) +ei

˜

θ +β

y₂e(˜θ)

hθ˜

#

dθ˜

˜

θ , θ∈[θ

i

m,θ) (Q-EC2’-i)

2

Assume thatθmi , i∈ {A,B}, is sufficiently big so that production is positive at this productivity

level. Then, from incentive compatibility (in the envelope condition form) and the requirement that production is non-decreasing in productivity, individual utility is strictly increasing onθ.

3

On the other hand, note thatW(θ,eA) andW(θ,eB) are two different functions, and not the same

We can now write the planner’s problem:4

max

c,y,εi,θim π

Z θA m

θ

W(θ)fA(θ)dθ+ (1−π)

Z θB m

θ

W(θ)fB(θ)dθ+π

Z θ θA m

W(θ,eA)fA(θ)dθ+

(1₋π)

Z θ θB m

W(θ,eB)fB(θ)dθ

s.t.

Z θ θ

[y1(θ) +βy2(θ)−c1(θ)−βc2(θ)]f(θ)dθ≥0 (Q-RC)

W(θ) =W(θ) +

Z θ θ

"

y1(˜θ)

˜

θ +β

y2(˜θ)

˜

θ

#

dθ˜

˜

θ , θ∈[θ,θ

i

m) (Q-EC1-i)

W(θ,ei) =W(θim) +

Z θ θi

m "

ye

1(˜θ) +ei

˜

θ +β

ye

2(˜θ)

hθ˜

#

dθ˜

˜

θ , θ∈[θ

i

m,θ] (Q-EC2’-i)

πFA(θAm) + (1−π)FB(θBm) =F(θm) (Q-CC)

fori_{∈ {}A,B_}.

Let λ be the Lagrange multiplier on (Q-RC), δi_(θ)f

i(θ) be the Lagrange multiplier

on (Q-EC1-i), γi(θ)fi(θ) the Lagrange multiplier on (Q-EC2’-i), and η the Lagrange

multiplier on on (Q-CC). The Lagrangian of the problem is found in LemmaA.2in the appendix.

Proposition 4.1. Let all regularity conditions necessary for the validity of the first-order approach and the cutoff rule be true. Suppose, without loss of generality, that

θA_m≤θ_mB. Then, the following is true of the optimal allocations are: (1) The Euler equation for consumption holds in each group.

• Ifθ < θA_m

U′(c1(θ)) =U′(c2(θ)) =

λf(θ)

f(θ) +πδA_(θ)f

A(θ) + (1−π)δB(θ)fB(θ)

• Ifθ_mA _≤θ < θB_m

U′(c1(θ)) =U′(c2(θ)) = λ

1 +δB_(θ)

if individual θis from group B, and

U′(ce₁(θ)) =U′(ce₂(θ)) = λ 1 +γA_(θ)

4

The notation c,y in the set of choice variables indicated in this version of the planner’s problem meansc={c1(θ),c2(θ),ce

22

if she is from group A.

• Ifθ_≥θB_m

U′(ce₁(θ)) =U′(ce₂(θ)) = λf(θ)

f(θ) +πγA_(θ)f

A(θ) + (1−π)γB(θ)fB(θ)

(2) One allocation of production in the second best optimum with quotas is

• θ < θA_m

y1(θ) =

1 2

θ(1 +β)U(c(θ))₋θW(θ)₋(1 +β)

Z θ θ

U(c(˜θ))dθ˜

and

y2(θ) = 1

2β

θ(1 +β)U(c1(θ))−θW(θ)−(1 +β)

Z θ θ

U(c(˜θ))dθ˜

.

• θA

m≤θ < θmB

– If in groupA:

ye₁(θ) = max

(

0, θ(1 +β)U(ce(θ))₋θ_mAW(θ_mA)₋(1 +β)

Z θ θA m

U(ce(˜θ))dθ˜₋eA−βy e

2(θ)

h

)

,

and

ye₂(θ) = min

(

ℓθh,h β

(

θ(1 +β)U(ce(θ))₋θ_mAW(θA_m)₋(1 +β)

Z θ θA m

U(ce(˜θ))dθ˜₋eA

))

.

– If in groupB:

y1(θ) = 1

2

θ(1 +β)U(c(θ))₋θW(θ)₋(1 +β)

Z θ θ

U(c(˜θ))dθ˜

,

and

y2(θ) = 1

2β

θ(1 +β)U(c(θ))₋θW(θ)₋(1 +β)

Z θ θ

U(c(˜θ))dθ˜

.

• θ_≥θB_m

– If in groupA:

ye₁(θ) = max

(

0, θ(1 +β)U(ce(θ))₋θ_mAW(θ_mA)₋(1 +β)

Z θ θA m

U(ce(˜θ))dθ˜₋eA−β

ye

2(θ)

h

)

,

and

ye₂(θ) = min

(

ℓθh,h β

(

θ(1 +β)U(ce(θ))₋θ_mAW(θA_m)₋(1 +β)

Z θ θA m

U(ce(˜θ))dθ˜₋eA

))

– If in groupB:

ye₁(θ) = max

(

0, θ(1 +β)U(ce(θ))₋θ_mBW(θ_mB)₋(1 +β)

Z θ θB m

U(ce(˜θ))dθ˜₋eB−β

ye

2(θ)

h

)

,

and

ye₂(θ) = min

(

ℓθh,h β

(

θ(1 +β)U(ce(θ))₋θ_mBW(θB_m)₋(1 +β)

Z θ θB m

U(ce(˜θ))dθ˜₋eB

))

.

Note that if there is bunching at the lower levels of productivity, we have

W(θ) = (1 +β)U(λ).

(3) There is bunching in the lower levels of productivity if at least one of the following is true:

• θ < θA

m and θ < π[1+δ

A_(θ)+∆

A(θ)]fA(θ)+(1−π)[1+δB(θ)+∆B(θ)]fB(θ)

λf(θ) ;

• θA_m≤θ < θ_mB and

(a) the individual is in groupAand θ < 1+γA(θ)+ΓA(θ)

hλ

(b) the individual is in groupB and θ < 1+δB(θ)+∆B(θ)

λ

• θB_m≤θand θ < π[1+γA(θ)+ΓA(θ)]fA(θ)+(1−π)[1+γB(θ)+ΓB(θ)]fB(θ)

hλf(θ) .

Proof. See AppendixA.2

Since the planner cannot discriminate the production allocations between groups and assuming thatℓ is sufficiently big,5 we must have, in the optimum,

θB_mW(θB_m) + (1 +β)

Z θ θB m

U(ce(˜θ))dθ˜+eB =θAmW(θmA) + (1 +β)

Z θ θA m

U(ce(˜θ))dθ˜+eA (4.1)

for allθ_≥max_{θA_m,θ_mB_}. This equation and the next proposition will help us investigate how the planner would optimally choose the effort thresholds (or cutoff points) between the two groups in the population.

Lemma 4.1. Suppose that the cutoff rule is valid, that is, Proposition3.2holds in each group and both W(θ) > W(θ,ei) and W(θ) < W(θ,ei) are true for i ∈ {A,B}. Then,

the optimal cutoff points in each group will be the solutions to equations (QRA) and (QRB), where

Z θ θA m

γA(θ)fA(θ)dθ=fA(θAm)

η+δA(θA_m)[W(θA_m)₋W(θ)] +λ[S(θ_mA)₋Se(θ_mA)]

Wθ(θAm)−Wθ(θAm,eA)

(QRA)

5

24

and

Z θ θB m

γB(θ)fB(θ)dθ=fB(θBm)

η+δB_(θB

m)[W(θBm)−W(θ)] +λ[S(θBm)−Se(θBm)]

Wθ(θBm)−Wθ(θBm,eB)

. (QRB)

Proof. See AppendixA.2

Lemma 4.1 explicits how the cutoff point and the effort threshold are related in a social optimum. These two equations, the optimal ‘quota rule’ in each group, together with restrictions (Q-EC3-A), (Q-EC3-B) and (4.1) yield a system of non-linear equations whose solution are the values we are looking for.

The first thing to notice is that the aforementioned system of equations has an extremely important normative feature: the equations do not depend on the proportions that different groups take of the population, π and 1₋π. This means that having educational quotas may be an optimal decision for a social planner, but only because it is a useful way to account for the information contained in an individuals tag, and not because one group takes a higher proportion of the population than does another. In other words, there should be no discrimination against – or in favor of – minoritiesjust because they are less numerous.

Equations (QRA) and (QRB) also elicit two different “effects” that play a significant role in the problem:

Z θ θi

m

γi(θ)fi(θ)dθ

| {z }

“Global effect”

= fi(θ

i

m){η+δi(θmi )[W(θim)−W(θ)] +λ[S(θmi )−Se(θmi )]}

Wθ(θmi )−Wθ(θmi ,ei)

| {z }

“Local effect”

, i_{∈ {}A,B_}.

The “global effect” (which is not actually global because it only encompasses individ-uals who invest in human capital) captures the idea that benefiting some group with a quota makes everyone in that group who invests in human capital better off. If the plan-ner wants to reduce the cutoff point in some group, he has to reduce the necessary effort for admittance in this same group, thus increasing the utility that a ‘infra-marginal’ individual gains when he invests in human capital and providing incentives for these individuals to study. However, because of incentive compatibility, in order to do this the planner has to increase the utility of all of the other individuals in that group who invest in human capital as well. The “global effect” is the aggregation of the shadow prices of incentive compatibility of individuals who study – in other words, the welfare effect of this whole process we have just described (the effect is the opposite in the not-benefited group).

investment decision has on the capacity, incentive and resource constraints. In quan-titative terms, the “local effect” measures the welfare gain of having and extra f(θi

m)

individuals not investing in human capital.

The fact that the optimal cutoff point equates these two effects brings some intuition to the result of lemma 4.1. Let us rephrase our question and ask: when is it optimal not to change the cutoff point in some group? The answer to this question naturally is: when there are no gains from doing it – the aggregate gains are compensated by the local distortions or vice-versa. In other words, the global effect is equal to the local effect.

Finally, let us now compare the two groups, and find out when is it optimal to have an educational quota.

Proposition 4.2. Suppose that Lemma4.1holds andη+λ[S(θm)−Se(θm)]6= 0. Then,

iffA(θm)6=fB(θm), it is optimal to have an educational quota.

Proof. See AppendixA.2

This result is quite surprising: if the skill distribution differs in a single point be-tween groups, and if this point happens to be θm, then the planner will implement an

educational quota. On the other extreme, if the skill distributions differ in every point except in θm, there is no quota! As surprising as this might be, the intuition behind it

is quite simple. Imagine that the social planner is thinking about benefiting group A

with a quota. IffA(θm) =fB(θm), then a small variation in the cutoff point designed to

let fA(θm) students from group A into college will, in turn, take fB(θm) students from

group B out of college. Since there is no explicit preference for a given group and no discrimination in the consumption and production allocations, the planner has gained nothing by introducing this infinitesimal quota. The only conclusion he can arrive at is to not implement a quota for any group. Note that, given Proposition4.2, we can define what we have been loosely referring assufficient information on a group’s tag as simply

fA(θm)6=fB(θm).

Alas, the system of equations that describes the optimal cutoffs in both groups does not have an analytic solution, so that there is little we can say about how θ_mA and θB_m

compare to each other, at this level of generality. Hence, although this result corroborates the point that the existence of affirmative action – or, more precisely, educational quotas – does not need to be justified solely by some exogenous preference for diversity at schools, there is still some important points that should be remembered.

The first of these points is: should a quota policy be redistributive, as are the ones we observe in practice? Drawing from economic intuition and previous insights in the literature of tagging and optimal taxation, we would expect that it should. However, these insights cannot immediately be generalized to the affirmative action problem, be-cause the introduction of a quota policy alters investment decisions of a non-negligible measure of individuals. Because of this fundamental feature of our problem, the planner has to deal with the “local effect”, to which we cannot unequivocally attribute a sign.

26

Possible Extensions

This chapter provides a discussion of some of the limitations of our model and points out a few potentially enlightening extensions to it. The first point we find worth mentioning is that we have a series of potential oversimplifications at work in our model. We assume, for example, that a person’s “productivity” can be represented by a single parameter θ

and that this parameter follows a well defined distribution over the population (which is public knowledge). However, people may be a lot more complicated than this, and could be better described by a series of values representing several dimensions of human behavior. Nevertheless, multidimensional screening has well known mathematical issues that we have chosen to avoid by collapsing all these dimensions into a single variable (which in turn makes it possible to talk about a “productivity distribution”). The technology of investment in human capital is also a potential source of oversimplification. It is a quite unrealistic hypothesis that individuals know exactly the amount of effort they need to produce – given their (one-dimensional) productivity – in order to be accepted and graduate from college. This entire process is probably subject to a lot of random turns of events. Having said that, we believe that adding random shocks to the model will bring us no further insight, but will undoubtedly be an extra source of complexity. Thus we leave it out of the model.

More concerning is the fact that an individual’s academic success may depend not only of his productivity and effort, but also on the quality of his previous education. In this case, and if high quality education is expensive, the cost of investing in human capital (relative effort) is a function of an individual’s after tax income, and thus an implicit function of the tax schedule. We do not address this question here, but it is, nonetheless, an interesting thought. A second interesting issue are the inter-generation consequences of affirmative action. Think of a situation where a person’s educational achievement is positively correlated with his parents’ educational achievement.1 In this case, by implementing “redistributive” quotas, a planner is not only increasing the level human capital investment of individuals in a given group in the present, but also in-creasing the level of human capital investment of the descendants of these individuals.

1

28

A pertinent question in this setting is if it is possible that by implementing an affirmative action policy for a finite period, we can permanently eliminate racial (or gender, etc) inequalities.

5.1

Inefficiency of quotas when there is discrimination on

production

A final issue is if having affirmative action policies would still be optimal if a social planner could tag groups in the income tax schedule as well. As we have seen, a quota policy is an instrument through which the planner can use the information he gets through the knowledge of a person’s origin (group A or group B). It is, additionally, the only instrument the planner has available to do so when he cannot discriminate the income tax schedule between groups. It is a reasonable thought that an income tax schedule is a more efficient way to use this information than a tax on effort is. In this case, a sensitive tax on effort could be just an imperfect way to mimic a group-sensitive tax on income. With a simple argument, we can show that this is exactly the case.

Suppose that the planner can discriminate completely between groups – that is, allocate a different consumption, production and effort to individuals with the same productivity but in different groups. Let y_te(θ) and ei be the optimal production and

effort allocations when the planner cannot discriminate production between groups, t_∈ {1,2_} and i_{∈ {}A,B_}. Also let ebe the minimum amount of effort needed to graduate from college. Define the group sensitive production allocation in the first period

y₁e(θ_|A) =y₁e(θ) +eA−e

and

y₁e(θ_|B) =y₁e(θ) +eB−e.

This ‘new’ allocation is incentive compatible, induces the same cutoff point in human capital investment and generates a higher aggregate product. It is, hence, a Pareto improvement.

The simple reasoning above shows that having quotas is always inefficient if the plan-ner could discriminate production (and therefore consumption) between groups. This strong result comes from the assumption that we made that a quota in universities only reallocates unproductive effort between groups, hence it does not not affect the human capital gains. However, since the effort fro acceptance into college has an unproductive component, it is actually a waste. If the social planner can instead induce a cutoff by varying production, he will always do so.

group get discounts on this fee (possibly raising the same fee for students from the non-targeted group). In many cases, this setting is plausible, especially when universities are private. In other cases, especially in some Latin American or European countries where a significant fraction of students tend to study in public universities, students do not generally pay any fee for their college studies, and this is not likely to change very soon.2 Our choice of not letting the planner discriminate production allocations between groups is in line with those cases.

2