Title:

Occupational Segregation and the (Mis)allocation of Talent∗

Author:

David Pothier

Address: DIW Berlin, Mohrenstrasse 58, 10117 Berlin Germany Email: [email protected]

Abstract:

This paper studies how occupational segregation affects the allocation of tal-

ent in a competitive labour market. A model of occupational choice is proposed in which heterogeneous workers must rely on their social contacts to acquire job vacancy information. While occupational segregation implies benefits in terms of job-finding probability, it also leads to allocative inefficiencies. Efficient and equilibrium outcomes differ due to a network externality that leads workers to segregate too little, and a pecuniary externality that leads workers to segregate too much. Which effect dominates depends on the elasticity of wages to changes in the degree of occupational segregation.

Keywords:

JEL Codes:

Occupational Choice, Social Networks, Allocation of Talent

J24, E24, D62

∗

I wish to thank Piero Gottardi, Fernando Vega-Redondo, Jan Eeckhout, Claudio Michelacci, participants of the EUI Microeconomics Working Group and the DIW Public Economics Cluster Seminar, as well as two anonymous referees for useful comments and suggestions. Financial support from the French Ministry of Education and Research is gratefully acknowledged.

1

I

Introduction

Occupational choice often not only depends on idiosyncratic characteristics (e.g. innate ability), but also on the occupations chosen by family, friends, and peers. Sociologists have coined the term ‘occupational segregation’ to refer to the sorting of individuals across occupations based on their social, religious, ethnic and/or gender identity. Such segregation is a prevalent feature of labour markets, and has been shown to be an important cause of wage and employment disparities between social groups.1 However, much less is known about how it affects the allocation of talent - i.e. the matching of skills to tasks - in the labour market. This paper studies the allocative implications of occupational segregation. The analysis centres around a theoretical model of occupational choice in which workers must rely on informal social networks to access job vacancy information. It builds on an idea originally put forward by Arrow (1998), who argued that the widespread use of referral networks in labour markets is an important cause of occupational segregation.2 A more recent paper by Buhai and van der Leij (2006) formalises Arrow’s original intuition, and shows that occupational segregation can be supported in equilibrium whenever individuals are disproportionately likely to form ties with other individuals belonging to the same social group - a phenomenon referred to as “homophilic inbreeding.”3 In order to study the allocative implications of occupational segregation, I extend their work by assuming that workers differ in terms of some publicly unobservable skill characteristic that determines both the cost of specialising in different occupations and their productivity if employed 1

See, for example, the studies by Albelda (1986) and King (1992). Empirical work dating back to Granovetter (1973) suggests that between 30% and 50% of all jobs are found using informal social networks. 3 There exists a wealth of empirical evidence demonstrating that such homophilic inbreeding is a widespread social phenomena. See for example the landmark study by McPherson, et al. (2001). 2

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by a firm. The assumption that workers must incur some specialisation costs before entering the labour market implies that they face a trade-off when choosing an occupation. On the one hand, they prefer to choose an occupation that is popular among their social contacts as this increases the probability that they find a job.4 On the other hand, they prefer to choose an occupation in which they are relatively more able, as this minimises the costs they must incur before searching for a job. The equilibrium degree of occupational segregation is determined by the relative magnitude of these two effects. The assumption that workers differ in terms of their productivity further implies that aggregate output depends on the allocation of talent across occupations. Inter alia, the model suggests that increases in the degree of segregation should lead to a decrease in equilibrium wages due to a misallocation of talent in the labour market. The key result of this paper pertains to the normative consequences of occupational segregation. Contrary to Buhai and van der Leij (2006), who find that occupational segregation is always desirable when workers are homogeneous in terms of their productivity, I show that the conditions needed for occupational segregation to be supported in equilibrium are generally not the same as those needed for it to be efficient. In particular, I show that workers’ reliance on social contacts to acquire job vacancy information generates a positive externality: by choosing to specialise in a given occupation, an individual increases the probability that his social contacts choosing the same occupation are successfully employed. If in addition to affecting his cost of specialisation, a worker’s skill-type also affects his productivity, workers’ reliance on social contacts to find a job generates a 4

This effect is consistent with recent empirical evidence showing that the use of social contacts in job search increases the probability of employment. For example, using a panel of local authority-level data from England between 1993 and 2003, Patacchini and Zenou (2012) find that increases in ethnic population density (meant to proxy for social networks and the transmission of job vacancy information) increases the ethnic employment rate.

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negative externality: when choosing an occupation, a worker does not internalise how his occupational choice decision affects the allocation of talent, and thereby aggregate labour productivity. This second externality results from firms’ inability to observe workers’ skill-type, implying that wage contracts cannot be written contingent on their productivity. This is consistent with the well-known result of Greenwald and Stiglitz (1986), who show that competitive equilibria are generically inefficient in economies with incomplete markets and asymmetric information. That being said, the inefficiency result I obtain also fundamentally depends on their being network effects in job search. In the absence of such homophilic networks, workers’ occupational choice decisions would be non-strategic and specialisation costs would drive them to choose the occupation in which they are relatively more able. In other words, pecuniary externalities would have no detrimental consequences for welfare, and equilibrium and efficient outcomes would coincide despite workers’ productivity being unobservable and markets incomplete. Lastly, I characterise the general conditions under which the equilibrium level of occupational segregation exceeds or falls short of the constrained efficient level. Whether decentralised markets exhibit too much or too little segregation is shown to be determined by how much wages react to changes in the level of segregation. In particular, given some fixed degree of network homophily, if the elasticity of wages is sufficiently high (low), then the magnitude of the pecuniary externality will be greater (smaller) than that of the network externality. Using data drawn from the Current Population Survey (CPS), I construct an index of racial segregation for the US. I then show how this data can be used to estimate a threshold wage elasticity to measure the relative magnitude of the two externalities mentioned above. The empirical results also support the key comparative static result of the model, as they indicate a significant negative relationship between real median weekly wages and occupational segregation in the US over the 2002-2014 period. 4

Related Literature In general, this paper contributes to the literature studying the interaction between informal social networks and competitive labour markets. An early and influential contribution is the paper by Montgomery (1991), who showed that homophilic social networks can be exploited by profit-maximising firms to costlessly screen job applicants. This spurred a large literature studying the dynamics of network-mediated job search.5 The job search technology used in the model below can be interpreted as a reduced-form representation of such dynamic job search processes. Topa (2001) studies both theoretically and empirically how individual employment status correlates with the employment status of workers’ social contacts. More recently, Dustmann, et al. (2011) and Galenianos (2014) develop labour market search models that explicitly incorporate network-based referrals. None of these papers, however, analyse how the ex ante investment decisions of workers is affected by their reliance on network-mediated recruitment channels. From a modelling perspective, this paper builds on existing models of segregation in competitive markets. The most closely related paper, as mentioned above, is the one by Buhai and van der Leij (2006). Their paper is itself based on the social interactions literature dating back to Schelling’s (1971) “tipping” model, and can be viewed as an adapted version of the framework used by Benabou (1993) to study how residential segregation affects workers’ human capital accumulation decisions. Moro and Norman (2004) also propose a theory of job segregation, but their model is based on statistical discrimination in the labour market rather than network effects in job search. As in this paper, they show that segregation can arise due to differences in human capital investments among ex ante identical groups. While Moro and Norman (2004) focus on vertically differentiated 5

See, for example, Calvo-Armengol (2004).

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occupations, this paper considers the case of horizontally differentiated occupations. Finally, this paper is also related to the extensive literature on the allocation of talent in labour markets. Moscarini (2001), for example, studies how search frictions and personal comparative advantage interact to determine the allocation of workers across occupations. His model, however, focuses on anonymous search mechanisms and he does not explicitly address the issue of network-mediated job search. The most closely related paper, in this regard, is the one by Bentolila, et al. (2010) which studies how social contacts in the labour market affect occupational mismatch. As in this paper, they find that social contacts imply both benefits (in terms of job-finding probability) and costs (in terms of labour productivity). Their paper, however, assumes an exogenous correlation between workers’ skill-types and the skill-type of their social contacts, and does not explore the conditions that determine the relative magnitude of the two externalities that distort the market allocation.

II

The Model

Workers I consider an economy populated by a continuum of risk-neutral workers, where N denotes the set of workers with measure normalised to two. Workers are ex ante heterogeneous and differ in terms of their skill-type, θ ∈ Θ = [0, 1]. Moreover, workers are equally divided into two social groups: reds (R) and greens (G). Skill-types are uniformly distributed in both groups so that θ ∼ U [0, 1] for all X ∈ {R, G}.6 This implies that no one group is ex ante predisposed to any particular occupation. Consistent with Loury’s (2002) axiom of 6

The uniform assumption is made to simplify the derivations. The qualitative nature of the results would be unchanged for any single-peaked symmetric distribution function.

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anti-essentialism, any occupational segregation that arises in equilibrium will thus be due to strategic considerations among the workers, rather than some presupposed productivity difference between individuals belonging to different social groups. There are two occupations for workers to choose from, φ ∈ Φ = {A, B}. Before entering the labour market, workers must choose to specialise in one of these two occupations. I assume that a worker cannot be hired by a firm unless he has specialised in an occupation.7 Depending on his chosen occupation, a worker’s skill-type determines his productivity if hired by a firm. Let zφ (θ) ∈ R+ denote the productivity of a type θ worker employed in occupation φ. Assumption 1. The productivity functions zφ (θ) for φ ∈ {A, B} satisfy the following conditions 1. Symmetry: zA (1 − θ) = zB (θ) 0 (θ) < 0 and z 0 (θ) > 0 2. Monotonicity: zA B

3. Weak Concavity: zφ00 (θ) ≤ 0 and zφ000 (θ) ≤ 0 ∀φ ∈ {A, B} These assumptions imply that workers located on the left hand side (near 0) of the unit interval are more productive in occupation A, while workers located on the right hand side (near 1) of the unit interval are more productive in occupation B. Workers located at the midpoint are equally productive in occupation A and B. A worker’s skill-type also determines the idiosyncratic cost he must incur in order to specialise in one of the two occupations. These costs are negatively correlated with productivity so that workers who find it relatively costly to specialise in a given occupation are also relatively less able at performing tasks if employed in that occupation. Let 7

The decision should be viewed as an investment in some observable and publicly recognised certificate needed for employment within a particularly industry (e.g. a law or architecture degree).

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cφ (θ) ∈ R+ denote the cost incurred by a type θ worker choosing to specialise in occupation φ. The restrictions imposed on the cost functions mirror closely those imposed on the productivity functions. Assumption 2. The cost functions cφ (θ) for φ ∈ {A, B} satisfy the following conditions 1. Symmetry: cA (1 − θ) = cB (θ) 2. Monotonicity: c0A (θ) > 0 and c0B (θ) < 0 3. Weak Convexity: c00φ (θ) ≥ 0 and c000 φ (θ) ≥ 0 ∀φ ∈ {A, B} These specialisation costs, and in particular the convexity assumption, are needed in order to guarantee the existence and uniqueness of an interior segregated equilibrium.

Social Contacts After having chosen an occupation, workers enter a competitive labour market. The labour market is subject to frictions in the sense that workers must rely on their social contacts to acquire job vacancy information. Metaphorically speaking, one can think of the labour market for each occupation as an island. Firms can costlessly open vacancies on each island and hire as many workers as they want. Isolated workers do not know the location of these ‘islands’ and are thus unaware of these vacancies, but the islands’ location can be revealed to them through information that is disseminated via their social contracts. Workers are more likely to receive information about vacancies when they have more contacts specialised in the same occupation. In particular, I assume social ties to workers specialised in a different occupation provide no job vacancy information whatsoever (to continue the island metaphor, information about the location of island A provides

8

no information about the location of island B ).8 I also assume that workers make their specialisation decisions before meeting their social contacts. As a result, workers’ occupational choice decisions only depend on their expectations about the group identity and occupational choices of their future social contacts.9 Social contacts exhibit an inbreeding bias, meaning that workers are disproportionately likely to have social ties with other workers belonging to the same social group. Denote by α ∈ (1/2, 1) the conditional probability that a randomly chosen worker socialises with another worker belonging to the same social group. Furthermore, let ηφX denote the expected measure of a worker’s contacts specialised in occupation φ when he belongs to group X. The probability that a worker is informed about a job vacancy is then given by q(ηφX ) ∈ (0, 1). In order to abstract from congestion effects in information transmission, I restrict attention to linear probability functions. Assumption 3. The job search function is linear so that q(ηφX ) = ηφX .

Firms Firms employ workers specialised in the two occupations and produce a homogeneous consumption good. I assume the two occupations to be essential and complements in production, so that firms must employ a positive measure of workers specialised in both occupations in order to produce. I normalise the price of the consumption good to unity, and let lφ ∈ R+ denote the labour supply of workers specialised in occupation φ. Firms 8

Intuitively, one should interpret the two occupations as being very different in terms of the skills they require. Building on the example provided above, job vacancies for architects are unlikely to be of interest to someone holding a law degree. 9 This is consistent with the interpretation whereby an individual’s social network is constituted of “weak” or “instrumental” ties. Granovetter (1973), among others, has shown that job vacancy information is more likely to be obtained from social connections made at university or in the labour market (and thus after workers have made some fixed investment in a career path), rather than from family or kinship ties.

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combine labour inputs using a neo-classical production technology with constant returnsto-scale, and total output is given by f (˜lA , ˜lB ), where ˜lφ ∈ R+ denotes labour supply in efficiency units. Assumption 4. The production function is symmetric and Cobb-Douglas such that β ˜1−β f (˜lA , ˜lB ) = ˜lA lB

with β = 1/2. The symmetry assumption is imposed in order to simplify the analysis and could be relaxed in order to study how occupational segregation affects wage and employment inequality between social groups.10 Since workers’ skill-types are publicly unobservable, the wage a worker receives if employed cannot be conditioned on his type. Moreover, I assume that wages and firms’ hiring decisions cannot be conditioned on workers’ social ‘colour.’11 The objective of the (representative) firm is then to choose labour demand schedules for both occupations in order to maximise its profits, as given by n o Π = max f (˜lA , ˜lB ) − wA lA − wB lB lA ,lB

(1)

In equilibrium, wages will be proportional to the average productivity of workers spe10

The assumption that the production technology is Cobb-Douglas is also made to simplify the model, but the results would be qualitatively similar given any symmetric production function satisfying the standard Inada conditions. 11 This assumption is not innocuous. Depending on their sorting behaviour, the expected productivity of workers in occupation φ need not be the same across social groups. If firms could condition workers’ pay on their social colour, this may make them less inclined to segregate in the first place. This assumption is made in order to focus on economies where, by law, firms are not allowed to condition wages and hiring decisions on workers’ gender, race or ethnicity.

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cialised in each occupation, implying that labour demand schedules must satisfy

wφ = E[zφ ]

∂f (˜lA , ˜lB ) , ∂ ˜lφ

∀φ ∈ {A, B}

(2)

where E[zφ ] denotes the expected productivity of workers employed in occupation φ.

Timing The timing of the model can be summarised as follows: • Stage 1: Workers choose to specialise in occupation φ and incur the cost cφ ∈ R+ . • Stage 2: Workers randomly meet to form an exogenous network of social connections (ηφX )φ∈{A,B} with group-bias parameter α ∈ (1/2, 1). • Stage 3: Workers find a job with probability q(ηφX ) ∈ (0, 1). Conditional on being employed in occupation φ, workers receive the wage wφ ∈ R+ . Employed workers use their wage earnings to purchase the consumption good produced by firms, while unemployed workers consume nothing.

III

Equilibrium and Welfare: Definitions

Equilibrium Let σ X (θ) ∈ [0, 1] denote the probability with which a type θ worker belonging to group R1 X chooses occupation A, and define σ X = 0 σ X (θ)dθ to be the total measure of workers in group X choosing occupation A. The payoff function of a type θ worker belonging to

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group X and choosing to specialise in occupation φ is given by      0 0 UφX σ X , σ X ; θ = q ηφX σ X , σ X wφ − cφ (θ)

(3)

where the first term equals a worker’s expected wage, while the second term equals the cost he must incur in order to specialise in occupation φ. Definition 1. A competitive equilibrium is defined as a specialisation strategy σ X (θ) for all X ∈ {R, G} and θ ∈ [0, 1], a labour demand schedule (lφ )φ∈{A,B} , and wages (wφ )φ∈{A,B} such that 1. Each worker chooses a specialisation strategy σ X (θ) to maximise his utility, taking wages and the occupational choice decision of other workers as given. 2. The representative firm chooses a labour demand schedule (lφ )φ∈{A,B} to maximise its profits, taking wages and the occupational choice decision of workers as given. 3. The labour market clears. Utility maximisation implies that workers’ specialisation decisions must satisfy

σ X (θ) = 0

if

∆φ U X (θ) < 0

σ X (θ) ∈ [0, 1]

if

∆φ U X (θ) = 0

σ X (θ) = 1

if

∆φ U X (θ) > 0

(4)

where ∆φ U X (θ) ≡ UAX (θ) − UBX (θ) denotes the payoff difference from choosing occupation A compared to occupation B.

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Definition 2. An equilibrium in threshold strategies is a competitive equilibrium such that the specialisation strategy of workers σ X (θ) satisfies the following condition ∃θˆX ∈ [0, 1]

∀X ∈ {R, G} :

σ X (θ) = 1

θ < θˆX

if

and

σ X (θ) = 0

if

θ > θˆX

The properties of the cost functions imply that restricting attention to equilibria in threshold strategies is without loss of generality. Lemma 1. All equilibria are necessarily in threshold strategies. Lemma 1 allows to derive explicit expressions for the (expected) measure of a workers’ social contacts specialised in each occupation 0 X ηA = αθˆX + (1 − α)θˆX ,

∀X ∈ {R, G}, X 6= X 0

(5)

0 X ηB = α(1 − θˆX ) + (1 − α)(1 − θˆX ),

∀X ∈ {R, G}, X 6= X 0

(6)

The effective labour supply in efficiency units can then be written as ˜lA =

X

X q(ηA )

X

zA (θ)dθ

(7)

0

X∈{R,G}

˜lB =

θˆX

Z

X q(ηB )

X∈{R,G}

Z

1

θˆX

zB (θ)dθ

(8)

In what follows, occupational segregation is defined as any deviation from the non-segregated threshold profile (θˆR , θˆG ) = (1/2, 1/2), whereby each worker chooses the occupation in which he is relatively more able. Complete occupational segregation is defined as one of the two corner solution threshold profiles: i.e. (θˆR , θˆG ) ∈ {(1, 0), (0, 1)}. Any intermediate threshold profile is referred to as partial occupational segregation.

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Social Welfare I focus on a welfare criterion based on utilitarian efficiency, rather than the more common notion of Pareto efficiency. The utilitarian welfare benchmark is a compelling one, especially if one is interested in comparing different outcomes from an ex ante perspective.12 Given that workers located at the extrema of the type space have a payoff advantage relative to those located in the middle of the type space, it makes sense to consider ex ante rankings of different potential equilibrium outcomes. Aggregate utilitarian welfare is given by W =

X

Z

1


σ X UAX (θ) + (1 − σ X )UBX (θ) dθ

(9)

X∈{R,G} 0

The task of a social planner seeking to maximise aggregate utilitarian welfare consists of specifying an allocation rule σiX (θ) : Θ → Φ for all X ∈ {R, G} and i ∈ N that allocates each worker of type θ ∈ [0, 1] to a specific occupation φ ∈ {A, B}. Using the definition of the payoff function given by equation (3), together with the labour supply equations (7)(8) and the assumption that the production function exhibits constant returns-to-scale, one can rewrite equation (9) in terms of threshold profiles as follows W (θˆR , θˆG ) = f (˜lA , ˜lB ) − C(θˆR , θˆG )

(10)

where C(θˆR , θˆG ) =

X X∈{R,G}

θˆX

Z

Z cA (θ)dθ +

0

!

1

θˆX

cB (θ)dθ

(11)

The planner’s problem is thus equivalent to maximising total output net of the aggregate specialisation costs. 12

See Harsanyi (1955).

14

Definition 3. A threshold profile (θˆR , θˆG ) ∈ [0, 1]2 is constrained efficient if it maximises aggregate utilitarian welfare (10) subject to the technological constraints (7)-(8). The definition of constrained efficiency implies that a social planner faces the same technological constraints as the competitive market and is required to match workers with vacancies using the network-mediated job search technology. The salient trade-off which characterises the constrained efficient allocation can be summarised as follows. On the one hand, the social planner would like to segregate workers belonging to different groups across occupations, as this increases the efficiency of the job search technology and thereby maximises aggregate employment. On the other hand, he must balance this against the increase in specialisation costs and the decrease in average labour productivity that arise when workers segregate. To gain a better intuition of this underlying trade-off, consider the two benchmark cases without inbreeding bias and with homogeneous skill-types, respectively. Absent any inbreeding bias, the social planner would have each worker choose the occupation in which he is most able, as this minimises total costs and allocative inefficiencies while leaving total output unchanged. If workers were homogeneous but the social network exhibited a positive inbreeding bias, the planner would instead have workers completely segregate across occupations, as this maximises aggregate employment while engendering no misallocation effects. In general, the symmetry properties imposed on the cost, productivity and production functions imply that the efficient allocation must also be symmetric. Lemma 2. The welfare maximising threshold profile (θˆR , θˆG ) is necessarily symmetric such that θˆR = 1 − θˆG .

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IV

Equilibrium and Welfare: Analysis

The objective of this section is to characterise the properties of the equilibrium and efficient allocations. Of particular interest are the properties of segregated equilibria, but notice that the non-segregated outcome can always be supported in equilibrium. This follows rather intuitively if one interprets workers’ occupational choice decisions as a classic coordination game: i.e. given that no worker chooses to segregate, the individual gains to choosing an occupation other than the one in which a worker holds a skill advantage are zero. This begs the question whether, and under what conditions, occupational segregation can arise in equilibrium? Moreover, how do these conditions compare to those needed for occupational segregation to be efficient?

Segregated Equilibrium Rewriting the optimality condition of workers (4) in terms of threshold strategies, any (interior) equilibrium threshold profile (θˆR , θˆG ) ∈ (0, 1)2 must satisfy the following system of indifference conditions 0 0 X X q(ηA )wA (θˆX , θˆX ) − cA (θˆX ) = q(ηB )wB (θˆX , θˆX ) − cB (θˆX ),

∀X ∈ {R, G}

(12)

For corner solutions, these conditions may hold as inequalities. In what follows, I restrict attention to symmetric allocations such that θˆR = 1− θˆG . This implies that labour supply, average productivity and wage rates will be equal across occupations. Note that there may exist other (asymmetric) equilibria in this economy. However, given Lemma 2, none of these asymmetric equilibria will ever be efficient. Since our goal is to understand the factors driving the divergence between equilibrium and efficient outcomes, it makes sense

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to restrict attention to symmetric allocations. Notice also that the symmetry assumption allows us to restrict attention to allocations belonging to the interval θˆ ∈ [1/2, 1] without loss of generality. Focusing on interior solutions, the indifference condition (12) becomes ˆ ≡ w(θ) ˆ (q(ηA ) − q(ηB )) − (cA (θ) ˆ − cB (θ)) ˆ =0 I(θ)

(13)

In order to guarantee the existence of an interior equilibrium, additional restrictions must be imposed on the parameter space. Assumption 5. The inbreeding bias parameter α is such that c0A (1/2) 1 cA (1) − cA (0) 1 < α < + R1 + R 1/2 2 2 2 zA (θ)dθ 0 zA (θ)dθ 0 Proposition 1. If Assumption 5 is satisfied, there exists a unique symmetric partially segregated equilibrium. For values of the inbreeding bias that exceed the upper threshold identified in Assumption 5, the equilibrium is characterised by complete occupational segregation. Conversely, for values of α less than the lower bound identified in Assumption 5, the equilibrium collapses to the non-segregated threshold profile. Corollary 1. Wages are strictly decreasing in the degree of occupational segregation. Corollary 1 implies that increases in the degree of segregation are associated with lower equilibrium wages for all workers, irrespective of their group identity. It comes about because occupational segregation leads to a poor matching of skills to tasks in the labour market (relative to the non-segregated case). This misallocation of talent reduces 17

the average productivity of labour across occupations, and this in turn has a depressing effect on the wages offered by profit-maximising firms.13 The magnitude of this effect depends on the variability of workers’ productivity across occupations. In the limiting case with homogeneous productivity, so that zA (θ) = zB (θ) for all θ ∈ [0, 1], there would be no productivity loss from misallocating talent and changes in the degree of segregation would have no effect on equilibrium wages.

Constrained Efficiency We now turn to study the normative properties of the segregated equilibrium identified in Proposition 1. By symmetry, one can rewrite the social welfare function (10) as follows Z W =

q(ηA )

θˆ

Z zA (θ)dθ + q(ηB )

0

!

1

θˆ

zB (θ)dθ

Z −2

θˆ

Z cA (θ)dθ +

0

!

1

θˆ

cB (θ)dθ

(14)

Again, to guarantee that the efficient allocation is interior, one must impose some restrictions on the parameter set. Assumption 6. The inbreeding bias parameter α is such that c0 (1/2) z 0 (1/2) 1 cA (1) − cA (0) (1 − α)zA (0) − αzA (1) 1 + A − A < α < + R1 + R1 2 2zA (1/2) 8zA (1/2) 2 2 0 zA (θ)dθ 0 zA (θ)dθ Under these conditions, the first-order condition of the social planner’s problem fully characterises the (interior) efficient allocation. As for the equilibrium, values of α that exceed the upper bound imply that the efficient allocation is characterised by complete occupational segregation, while values of α below the lower bound imply that social welfare 13

Note that this result does not depend on the assumption that workers’ skill-types are unobservable. Even if wages could be conditioned on types, increases in the degree of occupational segregation would still have a negative effect on (average) wages.

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is maximised at the non-segregated threshold profile. Proposition 2. If Assumption 6 is satisfied, then the social planner’s problem has a unique interior solution. Moreover, the first-order condition characterising the efficient allocation is given by ˆ ≡ w(θ) ˆ (q(ηA ) − q(ηB )) − (cA (θ) ˆ − cB (θ)) ˆ + XN + XP = 0 E(θ)

(15)

where 0 ˆ ˆ >0 X N = ηA (θ)(2θˆ − 1)w(θ)

and

ˆ θ) ˆ <0 X P = w0 (θ)l(

(16)

Evaluating the first-order condition of the social planner’s problem at the equilibrium allocation θˆEQ ∈ (0, 1) yields E(θˆEQ ) = I(θˆEQ ) +X N (θˆEQ ) + X P (θˆEQ ) ≶ 0 | {z }

(17)

=0

It follows from equation (17) that the relative magnitude of the two additional terms, X N and X P , determine how the equilibrium degree of occupational segregation compares to the efficient allocation. The term X N measures the positive externality generated by the 0 (θ) ˆ = (2α − 1), which measures the effect of job search technology. The key term is ηA

changes in the threshold θˆ on the job-finding probability of workers’ contacts specialised in occupation φ. This is the source of the network externality: a worker does not take into account how his occupational choice decision affects the job-finding probability of other workers in his social network choosing the same occupation. The term X P measures the negative pecuniary externality that arises when workers’

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ˆ productivity on the job varies as a function of their skill-type. The key term is w0 (θ), which measures the effect of changes in the threshold θˆ on wages. The fact that this effect is always negative follows from Corollary 1. This is the source of the pecuniary externality: a worker does not internalise how his occupational choice decision affect the allocation of talent in the labour market, and thereby equilibrium wages. It depends critically on two assumptions. The first is that workers’ skill-types cannot be observed by firms. If this were not the case, firms could offer a menu of wages which varies as a function of workers’ productivity. In such a complete-market setting the pecuniary externalities generated by workers’ occupational choice decisions would not imply an efficiency loss as wages would adjust so that workers receive their marginal product regardless of which occupation they choose. The second is that workers’ occupational choice decisions are strategic complements due to the presence of network effects in job search. If these were absent, and workers’ occupational choice decisions were non-strategic, the competitive equilibrium would be efficient. Even though workers’ skill-types would still be unobservable and markets incomplete, specialisation costs - which by assumption are negatively correlated with workers’ ability on the job - would drive them to choose the occupation in which they are relatively more able.

Discussion The normative results derived above mirror closely the conclusions reached by Bentolila et al. (2010), but it is important to underline in what ways they differ. They propose a searchand-matching model of the labour market in which workers can use their social contacts to find a job. As in this paper, they find that reliance on social contacts in job search imply both benefits (in terms of job-finding probability) and costs (in terms of labour

20

productivity). In equilibrium, the level of occupational mismatch exceeds the socially optimal level because workers do not internalise the adverse effect that a reduction in aggregate labour productivity has on vacancy creation. This inefficiency is closely related to the pecuniary externality identified by Acemoglu (1996): i.e. undirected search and ex post wage bargaining imply incomplete wage contracts which distort investment incentives. As pointed out by Acemoglu (1996), it depends on the fact that the bargaining protocol results in there being no direct mapping from workers’ productivity to the wage they receive in equilibrium. For example, the inefficiencies would disappear in an environment with ex ante wage posting and directed search. Contrary to Bentolila et al. (2010), wages in the model discussed above are set competitively on a Walrasian market. Nonetheless, the negative pecuniary externality identified above is conceptually very similar. It also depends on the fact that there is no direct mapping from productivity to wages. While in Bentolila et al. (2010) this market incompleteness arises due to the structure of the wage-setting mechanism, in this paper it is caused by the presence of asymmetric information in the labour market. That being said, the inefficiency result of Bentolila et al. (2010) fundamentally depends on the correlation between a worker’s skill-type and the skill-type of his social contacts, which they treat as exogenous. More importantly, their model remains silent about the conditions under which the positive externality generated by network effects in job search dominates, or is dominated by, the negative pecuniary externality implied by a misallocation of talent in the labour market. The next section addresses this particular issue in greater detail.

21

V

Threshold Wage Elasticity

Under what conditions does the equilibrium level of occupational segregation exceed or fall short of the constrained efficient level? Condition (17) allows for the derivation of a simple statistic that provides an answer to this question. ˆ α), where Corollary 2. Let Ew,θˆ denote the wage elasticity and define the threshold T (θ; ˆ θˆ w0 (θ) Ew,θˆ ≡ ˆ w(θ)

and

ˆ α) ≡ T (θ;

1−

α (2α−1)θˆ

(2θˆ − 1)

!−1 −1

Then, given some equilibrium level of occupational segregation θˆEQ ∈ (1/2, 1) and corresponding constrained efficient level θˆSP ∈ (1/2, 1), we have that    θˆEQ ≥ θˆSP

if Ew,θˆEQ ≥ T (θˆEQ ; α)

  θˆEQ < θˆSP

otherwise

(18)

This inequality relates the wage elasticity to the degree of homophilic inbreeding bias and the equilibrium level of occupational segregation. It states that if the elasticity is greater (smaller) than some threshold value, then the magnitude of the pecuniary externality will be greater (smaller) than the network externality. In principle, condition (2) can be estimated since all the variables it contains are observable. An econometrician armed with data on wages, sorting patterns and some proxy for the degree of homophily could thus use it (or preferably some analogue that controls for differences in group sizes and productivity) to determine the relative magnitude of the two externalities identified above. A common metric of occupational segregation is Duncan’s (1955) Index of Dissimilarity. This index measures the proportion of workers that would

22

need to change jobs in order for the demographic composition of each occupation to reflect the demographic composition of the population at large. For example, consider an economy consisting of J occupations and populated by X socio-demographic groups. The degree of group-specific segregation in occupation j is then defined as

Segjx = |(% of x in population) − (% of x in occupation j)|,

∀x ∈ X, j ∈ J

A composite index measuring the aggregate degree of occupational segregation in the economy as a whole can be obtained by summing across occupations and groups

Seg =

1XX P ropj Segjx 2 j∈J x∈X

where P ropj denotes the proportion of workers employed in occupation j. For an economy consisting of two-equally sized occupations and populated by two equally-sized social groups, as studied above, the threshold variable θˆ can be simply expressed in terms of this segregation index: θˆ = 1/2 + Seg. Using this expression, the threshold wage elasticity ˆ can be rewritten in terms of the segregation index as follows T (θ)  Ew,Seg ≡ Ew,θˆ

1 1− 2θˆ

 (19)

Data from the Current Population Survey (CPS) can be used to calculate segregation indices for the US. The CPS is a monthly household survey conducted for the Bureau of Labor Statistics (BLS) which provides a comprehensive body of data on employment, earnings and other demographic and labour force characteristics. In particular, it provides data on employment and earnings of full-time wage and salary workers in different occupations, decomposed by gender, race and ethnicity. Using this data, I calculated racial 23

Table 1: Wages and Segregation by Occupational Group, Average 2002-2014 Occupation Management/Professional Service Sales and Office Resource/Maintenance Production/Transportation Weighted Mean a

Wage ($) 819 367 492 555 470 572a

Racial Seg (%) 10.6 11.7 2.7 17.0 10.6 9.6

Gender Seg (%) 4.3 10.2 16.3 42.1 17.3 13.8

Seasonally-adjusted values.

and gender segregation indices over the 2002-2014 period across five major occupational groups.14,15 Summary statics are provided in Table 1. The data indicate that 9.6% of workers would need to change jobs in order for the racial/ethnic composition of each occupational group to reflect the racial/ethnic composition of the US workforce. For the gender-based index, this value rises to 13.8%. For illustrative purposes, consider a hypothetical economy consisting of two sectors and two groups that exhibits a degree of occupational segregation similar to that observed in the data. What would the threshold elasticity be for such an economy? Expressed in terms ˆ a segregation index of 10% implies a value of θˆ = 0.5+0.1 = 0.6. of the threshold variable θ, To calculate the threshold wage elasticity one also needs some reasonable value for the degree of homophily within workers’ social networks. Several empirical studies, including the paper by Currarini et al. (2009), have estimated the extent of homophilic inbreeding for various kinds of social networks. I set α = 0.85 based on their estimate of homophilic 14 The classification system for occupations is derived from the Standard Occupational Classification (SOC). Broad classifications are aggregated into minor groups, which are in turn aggregated into five major groups. 15 The racial segregation index is calculated based on the proportion of Blacks, Asians and Hispanics employed in each occupational group. The majority of the excluded demographic group consists of “white” workers or belonging to “some other race.” Note that the US Census divides race and ethnicity into two different categories, and that Hispanics and Latinos often classify themselves as “white” or belonging to “some other race.” Consequently, the excluded demographic group in the sample consists of all “white” or “other race” workers whose ethnicity is neither Hispanic nor Latino.

24

560

Median weekly wages ($) 570 580

590

Figure 1: Median Weekly Wages against Racial Segregation (Q1:2002-Q4:2014).

.085

.09

.095 .1 Racial segregation (index)

.105

inbreeding among white students in an American high school.16 This yields a threshold value of T (0.6, 0.85) = −0.16. Using equation (19), this corresponds to a threshold wage elasticity (expressed in terms of the segregation index) equal to -0.03. How does this value compare to elasticities observed in the data? Obtaining a reliable empirical estimate of the elasticity of wages to changes in the degree of occupational segregation, especially at the sectoral level, goes beyond the scope of this paper. The CPS data summarised above, however, can be used to obtain a rough indication of the order of magnitude involved. Using quarterly data from 2002 to 2014, Figure 1 plots real median weekly wages (seasonally adjusted and averaged across occupational groups) 16

Using friendship data from an American high school obtained from the Add Health dataset, Currarini et al. (2009) find that the degree of homophily among white students, as measured by the Coleman index, equals 0.69. As white students constitute 51% of their sample, this implies a value of α = 0.51 + 0.69(1 − 0.51) = 0.85.

25

against the racial-segregation index calculated above. The observed negative correlation between segregation and wages is consistent with the key comparative static of the model (see Corollary 1).17 In terms of magnitude, the implied wage elasticity can be calculated by regressing the logarithm of median real wages on the logarithm of the segregation index. The regression results yield an estimated wage elasticity equal to -0.08.18,19 While this coefficient should be interpreted with caution, it suggests that the threshold elasticities implied by the model are roughly of the same order of magnitude as those found in the data. Returning to the hypothetical economy described above, the model suggests that the pecuniary externality would dominate the network externality if the wage elasticity corresponded to that observed in the data (since | − .08| > | − 0.03|). This implies that the equilibrium level of occupational segregation in such an economy would be inefficiently high, and that aggregate productivity and output could be increased if workers were less segregated.

VI

Conclusion

This paper studies how occupational segregation affects the allocation of talent in a competitive labour market. A model of occupational choice was proposed in which heteroge17 The outliers in Figure 1 (real median weekly wages in excess of $580) correspond to the quarters immediately following the 2008/2009 recession: i.e. 2009Q1 through 2010Q4. 18 The regression model is given by:

log(M edianW aget ) = β0 + β1 log(SegIndext ) + t The estimated coefficient of the regression model is β1 = −.08 with a robust standard error equal to 0.019, implying statistical significance at the 1% level. Overall, the magnitude of the effect remains small, with a 1% increase of the segregation index being associated with a $4.8 decrease in real median weekly wages. Based on the R2 , changes in occupational segregation can explain roughly 14% of the observed variation in median weekly wages over the sample period. 19 The same regression model was estimated using the gender-segregation index, rather than racialsegregation index. The results are similar, with an estimated coefficient equalling -.08 and a robust standard error equal to 0.041, implying statistical significance at the 5% level.

26

neous workers must rely on their social contacts to acquire job vacancy information. In this environment, network effects in job search lead to occupational segregation arising in equilibrium. The analysis showed that the equilibrium level of segregation does not generally coincide with the constrained efficient level. Inefficiencies arise because workers do not internalise how their individual occupational choice decisions affect either: (i) the job finding probability of other individuals belonging to their social network, or (ii) the average productivity of labour across occupations. Which of these two effects dominates depends on the elasticity of wages to changes in the degree of segregation: if the wage elasticity is sufficiently high (low), then the magnitude of the pecuniary externality will be greater (smaller) than that of the network externality. As a clarifying remark, this paper does not wish to argue that gender-based or racebased discrimination is not an important cause of observed occupational segregation. Rather, the model developed here seeks to highlight another channel through which occupational segregation can arise in competitive labour markets. To this extent, it serves to underline the view that even in societies in which the level of overt discrimination is on the wane (due to either broad societal changes or more specific political causes), there is reason to believe that occupational segregation will persist as a salient feature of labour markets. What is more, the normative results obtained above indicate that in economic environments in which (i) social contacts play an important role in allocating workers to vacancies, and (ii) workers’ productivity is private information, occupational segregation may lead to an inefficient allocation of talent in the labour market. This suggests that some form of policy intervention in labour markets may be justified, even though a detailed discussion of the specific features of such policy intervention lies beyond the scope of this paper.

27

References Acemoglu, D., “A microfoundation for social increasing returns in human capital accumulation,” The Quarterly Journal of Economics, 1996, 111 (3), 779–804. Albelda, R.P., “Occupational segregation by race and gender, 1958-1981,” Industrial and Labor Relations Review, 1986, pp. 404–411. Arrow, K.J., “What has economics to say about racial discrimination?,” The Journal of Economic Perspectives, 1998, 12 (2), 91–100. Benabou, R., “Workings of a city: location, education, and production,” The Quarterly Journal of Economics, 1993, 108 (3), 619–652. Bentolila, S., C. Michelacci, and J. Suarez, “Social contacts and occupational choice,” Economica, 2010, 77 (305), 20–45. Buhai, I.S. and M.J. Leij, “A social network analysis of occupational segregation,” 2006. Tinbergen Institute Discussion Paper. Calv´ o-Armengol, A., “Job contact networks,” Journal of Economic Theory, 2004, 115 (1), 191–206. Currarini, Sergio, Matthew O Jackson, and Paolo Pin, “An economic model of friendship: Homophily, minorities, and segregation,” Econometrica, 2009, 77 (4), 1003–1045. Duncan, O.D. and B. Duncan, “A methodological analysis of segregation indexes,” American Sociological Review, 1955, 20 (2), 210–217.

28

Dustmann, C., A. Glitz, and U. Sch¨ onberg, “Referral-based job search networks,” 2011. IZA Discussion Paper, No. 5777. Galenianos, Manolis, “Hiring through referrals,” Journal of Economic Theory, 2014. Granovetter, M.S., “The strength of weak ties,” American Journal of Sociology, 1973, pp. 1360–1380. Greenwald, B.C. and J.E. Stiglitz, “Externalities in economies with imperfect information and incomplete markets,” The Quarterly Journal of Economics, 1986, 101 (2), 229–264. Harsanyi, J.C., “Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility,” The Journal of Political Economy, 1955, 63 (4), 309–321. King, M.C., “Occupational Segregation by Race and Sex, 1940-88.,” Monthly Labor Review, 1992, 115 (4), 30–36. Loury, G.C., The Anatomy of Racial Inequality, Harvard University Press, 2002. McPherson, M., L. Smith-Lovin, and J.M. Cook, “Birds of a feather: Homophily in social networks,” Annual Review of Sociology, 2001, pp. 415–444. Montgomery, J.D., “Social networks and labor-market outcomes: Toward an economic analysis,” The American Economic Review, 1991, 81 (5), 1408–1418. Moro, Andrea and Peter Norman, “A general equilibrium model of statistical discrimination,” Journal of Economic Theory, 2004, 114 (1), 1–30. Moscarini, Giuseppe, “Excess worker reallocation,” The Review of Economic Studies, 2001, 68 (3), 593–612. 29

Patacchini, E. and Y. Zenou, “Ethnic networks and employment outcomes,” Regional Science and Urban Economics, 2012. Schelling, T.C., “Dynamic models of segregation,” Journal of Mathematical Sociology, 1971, 1 (2), 143–186. Topa, G., “Social interactions, local spillovers and unemployment,” The Review of Economic Studies, 2001, 68 (2), 261–295.

A

Appendix: Proofs

Proof of Lemma 1 Since the problem is symmetric for workers of both groups, it suffices to show that the claim holds for workers of one group X ∈ {R, G}. For notational simplicity, denote the difference in expected wages across X X )wB . Consider first the case of a candidate equilibrium strategy )wA −q(ηB occupations by ∆φ E[w]X = q(ηA 0

X X X X (θ0 ) (θ0 ) < UB (θ) for some θ ∈ Θ and UA (θ) > UB profile (σ X (θ), σ X (θ))θ∈Θ with X 6= X 0 such that UA

for some θ0 ∈ Θ. Then, utility maximisation implies that ∆φ cφ (θ) < ∆φ E[w]X < ∆φ cφ (θ0 ). Since cA (θ) is increasing in θ and cB (θ) is decreasing in θ, θ0 > θ. Moreover, by continuity and monotonicity of the X X ¯ X ¯ (θ) is (θ). Since UA (θ) = UB functions cA (θ) and cB (θ) there must exist a unique θ¯ ∈ Θ such that UA X X X X X (θ) (θ) > UB (θ) ∀θ > θ¯ while UA (θ) < UB (θ) is increasing in θ, it must be that UA decreasing in θ while UB

¯ Thus, in equilibrium σ X (θ) = 1 ∀θ < θ¯ and σ X (θ) = 0 ∀θ > θ. ¯ But this is just the definition of ∀θ < θ. a threshold strategy. The argument easily extends to the case of candidate equilibrium strategy profiles X X X X (θ) ∀θ ∈ Θ, so that (θ) > UB (θ) ∀θ ∈ Θ, so that σ X (θ) = 1 ∀θ ∈ Θ, or UA (θ) < UB where either UA

σ X (θ) = 0 ∀θ ∈ Θ. 

Proof of Lemma 2 The claim follows from the symmetry assumptions imposed on the cost, productivity and production functions. Since the social welfare function is additively separable, I prove the claim by demonstrating that the production function f (·) reaches its maximum value and the aggregate cost function C(·) reaches

30

its minimum value on the line defined by θˆR = 1 − θˆG . Without loss of generality, one can write θˆR = x + θˆG , where x ∈ [−1, 1]. Substituting this condition into the cost functions yields C(θˆR , θˆG ) = C(θˆG ; x). Differentiating this function with respect to θˆG and using the symmetry assumption cA (θ) = cB (1 − θ) yields 

   cA (θˆG ) − cA (1 − θˆG ) + cA (x + θˆG ) − cA (1 − x − θˆG ) = 0

It is easily verified that this condition holds if θˆG = (1 − x)/2. To verify that this is indeed a unique minimum, notice that dC 2 (θˆG ; x) = c0A (θˆG ) + c0A (1 − θˆG ) + c0A (x + θˆG ) + c0A (1 − x − θˆG ) > 0 dθˆG2 This inequality follows from the assumption that the function cA (·) is monotonically increasing, implying that the cost function C(θˆG ; x) is globally convex. Turning now to the production function f (·), notice that the symmetry assumption implies that the maximum must satisfy the following condition ∂f (˜ lA , ˜ lB ) ∂f (˜ lA , ˜ lB ) = ⇔˜ lA = ˜ lB ˜ ∂ lA ∂˜ lB Using equations (7)-(8), together with θˆR = x + θˆG , this last equality implies q(θˆG + αx)

Z

θˆG +x

zA (θ)dθ + q(θˆG + (1 − α)x)

zB (θ)dθ = 0

0

q(1 − θˆG − αx)

θˆG

Z

Z

1

zA (θ)dθ + q(1 − θˆG − (1 − α)x)

θˆG +x

Z

1

zB (θ)dθ θˆG

It can be verified that this condition is again satisfied if and only if θˆG = (1 − x)/2. The final step of the proof simply requires to notice that since x ∈ [−1, 1], the locus of points defined by the condition θˆG = (1 − x)/2 is nothing more than the line θˆR = 1 − θˆG . 

31

Proof of Proposition 1 Given equations (7) and (8), the average ability of workers across occupations is defined as

E[zA ] =

E[zB ] =

   0

if

θˆX = 0

∀X ∈ {R, G} (A.1)

ˆX X Rθ zA (θ)dθ X∈{R,G} q(ηA ) 0 P q(η X )θˆX

P

     0

X∈{R,G}

otherwise

A

if X R1 X∈{R,G} q(ηB ) θ ˆX zB (θ)dθ P X ˆX X∈{R,G} q(ηB )(1−θ )

θˆX = 1

∀X ∈ {R, G} (A.2)

P

 

otherwise

Using the symmetry restriction, equations (5)-(6) become ηA = (2α − 1)θˆ + (1 − α),

and

ηB = (1 − 2α)θˆ + α

As the job search function is linear, the indifference condition (13) can be rewritten as     ˆ ≡ w(θ)q (2α − 1)(2θˆ − 1) − cA (θ) ˆ − cB (θ) ˆ =0 I(θ) Combining equation (2) and (A.1)-(A.2) yields

ˆ ≡ w(θ)

R θˆ R1 1 q(ηA ) 0 zA (θ)dθ + q(ηB ) θˆ zB (θ)dθ 1 E[z] = , ˆ 2 2 q(ηA )θˆ + q(ηB )(1 − θ)

∀φ ∈ {A, B}

Notice that since the non-segregated threshold profile is always an equilibrium, I(1/2) = 0. By Assumption 5, it must also be that I(1) < 0. Differentiating the indifference condition with respect to θˆ yields     ˆ = w0 (θ)q ˆ ˆ 0 (·) − c0A (θ) ˆ − c0B (θ) ˆ I 0 (θ) (2α − 1)(2θˆ − 1) + 2(2α − 1)w(θ)q Again, from Assumption 5 it must be that I 0 (1/2) > 0. To prove the claim, it thus suffices to show that ˆ is globally concave. Differentiating again with respect to θˆ yields the function I(θ)     ˆ = w00 (θ)q ˆ ˆ 0 (·) − c00A (θ) ˆ − c00B (θ) ˆ <0 I 00 (θ) (2α − 1)(2θˆ − 1) + 4(2α − 1)w0 (θ)q ˆ (see the proof of Corollary 1 where the inequality follows from the properties of the wage function w(θ) below) and the assumption that the cost functions are weakly convex. 

32

Proof of Corollary 1 ˆ as specified by equation (2) and the definition of average productivity E[z] Given the wage function w(θ) given by equations (A.1)-(A.2), one obtains ˆ <0⇔˜ ˆ θ) ˆ < l0 (θ) ˆ˜ ˆ w0 (θ) l0 (θ)l( l(θ)

where ˆ = (q(ηA ) − q(ηB )) + q 0 (·)(2α − 1)(2θˆ − 1) l0 (θ) and ˜ ˆ = (q(ηA )zA (θ) ˆ − q(ηB )zB (θ)) ˆ + q 0 (·)(2α − 1) l0 (θ)

θˆ

Z

!

1

Z zA (θ)dθ −

zB (θ)dθ θˆ

0

ˆ˜ ˆ = A1 (θ) ˆ + B1 (θ) ˆ where For notational simplicity, I write l0 (θ) l(θ) ˆ = q(ηA ) A1 (θ)

2

θˆ

Z

zA (θ)dθ − q(ηB )

2

Z

1

!

1

Z zA (θ)dθ −

zB (θ)dθ − q(ηA )q(ηB ) θˆ

0

θˆ

Z

zB (θ)dθ θˆ

0

and 0

ˆ = q (·)(2α − 1)(θˆ − (1 − θ)) ˆ B1 (θ) q(ηA )

θˆ

Z

!

1

Z zA (θ)dθ + q(ηB )

zB (θ)dθ θˆ

0

ˆ θ) ˆ = A2 (θ) ˆ + B2 (θ) ˆ where Similarly, ˜ l0 (θ)l(   ˆ = q(ηA )2 zA (θ) ˆ θˆ − q(ηB )2 zB (θ)(1 ˆ − θ) ˆ − q(ηA )q(ηB ) zB (θ) ˆ θˆ − zA (θ)(1 ˆ − θ) ˆ A2 (θ) and θˆ

  Z ˆ = q 0 (·)(2α − 1) q(ηA )θˆ + q(ηB )(1 − θ) ˆ B2 (θ)

Z zA (θ)dθ −

0

!

1

zB (θ)dθ θˆ

ˆ > B2 (θ). ˆ After simplifying, this condition implies I begin by demonstrating that B1 (θ) θˆ

Z

1

ˆ zB (θ)dθ > (1 − θ)

θˆ

θˆ

Z

zA (θ)dθ 0

Rearranging and using the fact that the productivity functions are symmetric yields 2θˆ − 1 1 − θˆ

!

R θˆ ˆ ˆ zA (θ)dθ zA (1 − θ) > R 1−θˆ 1−θ ˆ zA (1 − θ) z (θ)dθ 0

33

A

where the inequality follows from the fact that zA (θ) is monotonically decreasing. I next show that ˆ > A2 (θ). ˆ Since zA (θ) is monotonically decreasing, it must be that A1 (θ) θˆ

Z

! ˆ θˆ − zA (θ)dθ − zA (θ)

1−θˆ

Z

! ˆ − θ) ˆ zA (θ)dθ − zA (1 − θ)(1

>0

0

0

and Z

θˆ

ˆ θˆ − zA (θ)(1 ˆ − θ) ˆ <0 zA (θ)dθ − zA (1 − θ)

1−θˆ

Since ηA ≥ ηB for all values of θˆ ≥ 1/2 (see the proof of Proposition 1), this completes the proof. 

Proof of Proposition 2 Differentiating the social welfare function (14) with respect to θˆ yields the following first-order condition θˆ

Z   ˆ ≡ q(ηA )zA (θ) ˆ − q(ηB )zB (θ) ˆ + q 0 (·)(2α − 1) E(θ)

Z zA (θ)dθ −

0

!

1

zB (θ)dθ

  ˆ − cB (θ) ˆ − 2 cA (θ)

θˆ

Begin by noticing that E(1/2) = 0. By Assumption 6, one must have E(1) < 0. Differentiating the first-order condition with respect to θˆ yields       0 ˆ 0 ˆ ˆ = q(ηA )zA ˆ + zB (θ) ˆ − 2 c0A (θ) ˆ − c0B (θ) ˆ E 0 (θ) (θ) − q(ηB )zB (θ) + 2q 0 (·)(2α − 1) zA (θ) Again, from Assumption 6, one must have E 0 (1/2) > 0. To prove the first part of the claim, it thus suffices ˆ is globally concave. Differentiating again with respect to θˆ yields to show that the function E(θ)       00 ˆ 00 ˆ 0 ˆ 0 ˆ ˆ = q(ηA )zA ˆ − c00B (θ) ˆ <0 E 00 (θ) (θ) − q(ηB )zB (θ) + 3q 0 (·)(2α − 1) zA (θ) + zB (θ) − 2 c00A (θ) where the inequality follows from the weak concavity of the productivity functions and the weak convexity of the cost functions. To derive the desired expression for the social planner’s first-order condition, notice that the derivative of the wage function (2) satisfies ˆ θ) ˆ = w0 (θ)l(

1 ˜0 ˆ ˆ 0 (θ) ˆ l (θ) − w(θ)l 2

34

From the proof of Corollary 1, one can rewrite this equation as ˆ θ) ˆ =1 w (θ)l( 2 0

ˆ − q(ηB )zB (θ)) ˆ + q 0 (·)(2α − 1) (q(ηA )zA (θ)

θˆ

Z

Z zA (θ)dθ −

0

!!

1

zB (θ)dθ

...

θˆ

  ˆ (q(ηA ) − q(ηB )) + q 0 (·)(2α − 1)(2θˆ − 1) − w(θ) Using the first-order condition of the social planner’s problem derived above, this expression becomes   ˆ ˆ θ) ˆ = E(θ) + (cA (θ) ˆ − cB (θ)) ˆ − w(θ) ˆ (q(ηA ) − q(ηB )) − q 0 (·)(2α − 1)(2θˆ − 1) w0 (θ)l( 2 ˆ and simplifying yields Solving this equation for E(θ) 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ w(θ)(q(η A ) − q(ηB )) − (cA (θ) − cB (θ)) + (2α − 1)(2θ − 1)w(θ) + w (θ)l(θ) = 0

35