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Economic Models of Discrimination

Sendhil Mullainathan

Economics 1035

Fall 2007

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Overview

• Describe a simple labor model

• Incorporate discrimination into the model

• Use this model to interpret audit studies

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Setup

• Production Firm:– Employs E workers– Suppose all workers earn the same amount– Quantity Produced is a function of the number

of workers: q=f(E)

– Pays wage w– Therefore Profits are:

p·q – wE = p·f(E) - w·E

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Optimization

• Again, Profits are:

p·f(E) - w·E

• First order condition for optimal E, E* p·f’(E*) = w

• Interpretation?– Firms hire workers until their marginal product

(the extra units they would produce) equals their cost

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Two types of workers

• Now suppose there are two types of workers A and D, advantaged and disadvantaged– Suppose the market pays the same wage for

both workers– A and D are substitutable

• Firms Maximize Profits: p·f(A+D) - w·(A+D)

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Optimum

• Again total employment is such that:p·f’(E*)=w

• How many A and D workers will a given firm hire?– The model does not say. They will be indifferent. – A firm could hire all of one or all of another.

• How many A and D workers would the market as a whole hire? – Determined by their labor supply curve. – But there is nothing here to encourage discrimination

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Some room for discrimination

• Suppose now that the market wage is different: wA and wD

• What would happen now?

• Optimization: – wA < wD Hire all A

– wA > wD Hire all D

• Why no discrimination?– Firms have no motive.

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How to model discrimination

• Possibilities:

– Firms only want to hire A. What’s the problem here?• No ability to make tradeoffs. Economics is most useful when

there are smooth tradeoffs

– Discriminatory firms have a “preference” for hiring A. How to model?

• Easy way of doing it: Include a cost of hiring D. • Profits:

p·f(A+D)- wA·A - wD·D – d·D• Here d is the strength of the firm’s discriminatory preference

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Optimization

• What will the firm do? Recall profit function:p·f(A+D)- wA·A - wD·D – d·D

• Depends on wages:– wA < wD · (1+d) Hire only As– wA > wD · (1+d) Hire only Ds– wA = wD · (1+d) Indifferent

• Even discriminatory firms hire D’s. Why?– If they are sufficiently cheap. – The required discount for D rises with d– But will they hire the same number?

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Observations

• Firms that hire all A’s– Lose money because they are paying for

more expensive workers– Inefficient scale

• Firms that hire all Ds – Still can lose money if d > 0. Inefficient scale.– They hire too few Ds. Why?

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Discriminatory Firm’s Profits

• Questions– Why are profits initially falling?– Why a discrete drop?– What is this point?– Why is it flat

thereafter?

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• Why are profits falling?– A firm with greater discrimination is inefficiently hiring

• Why a discrete drop at a point?– At d = wA/wD – 1, the firm is indifferent between D and

A. When it switches to A’s, profit falls.– But why is it discrete? Compensating differential

• Why is it flat?– Once hiring all A workers, greater d doesn’t affect

behaior

• What happens to high d employers?– They earn less profits

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Profits as a function of wages

• Questions– Why are profits falling initially?– Why a discrete drop?– What is this point?– Why is it flat

thereafter?

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• Why are profits falling?– When hiring all D workers, as their wage rises, profit

falls.

• Why a discrete drop at a point?– This is the point at which wA =wD (1+d). So the firm is

indifferent between the two workers. – When it switches to As, profit falls.– But why is it discrete? Compensating differential

• Why is it flat?– Once hiring all A workers, the wage of D does not

matter

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Equilibrium

• There will be a gap.• Is this always true? • What does this

graph assume?

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Equilibrium

• There is still a gap

• Is this always true?

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Equilibrium

• No longer a gap

• How many d=0 firms are needed?

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Key Insight

• Wage differential is determined by the nature of the marginal firm, not the average firm

• What does this mean?– All the D workers sorted to firms with low d,

the non-discriminators.– If there are enough them, there will be no

wage impact.

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Other observations

• There will be segregation• Profit of discriminators will be the same as non-

discriminators if there are enough non-discriminators

• If there are not, discriminators will pay a “price”• What should happen to them in the long-run?

– They will not be able to compete with non-discriminators and should leave the industry.

• Assumes there are enough non-discriminators to run the firms

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Critique of Audit Studies

• They only measure average discrimination.

• Not what happens in wages.

• Responses?– Market sorting is not perfect. – Job search is an inefficient process.

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Statistical DiscriminationA Different Model

• Employers are profit maximizing

• Workers have productivity p.

• Firms would like to hire any worker and pay wage w=p.

• But productivity is uncertain. – They see a signal s.

• So they will pay w=E[p|s]

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Race might matter

• Case 1: Suppose that average productivity of D is lower than A.– Then they will pay E[p|s,D] or E[p|s,A]– So even with the same signal, D’s can get

paid less

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Race might matter

• Case 1: Suppose that average productivity of D is lower than A.– Then they will pay E[p|s,D] or E[p|s,A]– So even with the same signal, D’s can get

paid less

• Case 2: Average productivity is the same but Firms “understand” s less for D’s– So will put less weight on s signal for D. – Key insight: Low performing D will do better

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Testing for these models

• How would you test for these models?• As information increases, gap decreases

– Any evidence you’ve seen– Recall resume audit study. What was found there?

Increasing gap– Altonji-Pierre: Race gap shortest at entry into job

• How else to test?– What if you could vary how much information is seen?

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Conclusion

• Simple economic models of discrimination depend on preference– Employers, workers, customers– Basic insight is sorting to lessen impact of

discrimination– Discriminators can pay a tax

• Statistical discrimination models emphasize using group membership as a signal