X. The economics of superstars. Motivation Some workers seem to earn very large wages: sports, movies, top managers, etc These rents seem associated with.

X. The economics of superstars

Motivation

• Some workers seem to earn very large wages: sports, movies, top managers, etc

• These rents seem associated with their ability to spread their talent over a large market

• That ability in turn depends on the technology that is used

Key ingredients:

• Labor is not a homogeneous input but a « quality » input– I cannot replace 1 good manager by 2 good ones– Each type of individual is a specific factor

• Higher quality workers cover a larger market, which acts as a multiplier effect on their wages; the lower the decreasing returns, the stronger that effect– Television is more inegalitarian than theatre

A simple model of managerial skills

• Continuum of workers of quality q

• Firms produce a homogeneous good sold at price p

• A firm needs exactly one worker (the manager)

• Managerial quality improves the firm’s productivity at all output levels

• Cost of producing y is c(y)/q

The wage schedule

• I cannot purchase managerial quality ina market for a homogeneous q

• Instead, I must hire an individual of a given type q

• As many different labor inputs as individual types

• Distribution of wages is characterized by wage schedule ω(q), rather than ωq

Determination of the wage schedule

Properties of the wage schedule

• Each individual earns a Ricardian rent on q

• Wages grow with skills

• Returns to skills related to returns to scale at the firm level

The basic mechanism:

• The steeper the marginal cost curve, the smaller the response of the firm’s output to an increment in managerial quality

• The smaller the difference in the scale of operations between two managers of different quality

• The lower the wage differential between managers

A specification

•Lower γ greater replicability more convex and inegalitarian wage schedule

q

ω

Figure 7.1: Impact of a reduction in γ on the distribution of income

How general are those results?

• Returns to skills fall with the local elasticity of the cost function

• But that local elasticity is computed around a different point if c.f. shifts

• One can construct examples where greater replicability does not affect the distribution of income

A counter-example

• The increase in size induced by a fall in φ in itself increases the local elasticity of costs

Extension: Occupational choice

• Suppose people elect between becoming managers (ω(q)) and workers (w)

• If worker, I work in an alternative ‘commodity’ sector

• Assume commodity price = 1, and worker productivity there = w– w is pinned down

• p is now endogenous and determined by equality of supply and demand

q

Ln ω(q)

Figure 7.2: Income distribution and the allocation of talent

w

Workers Managers

How is the critical level determined?

• Indifference of critical workers yields negative relationship between p and q*

• Demand for the good yields a positive relationship

q*

p

Supply of commodities

Demand for commodities

Figure 7.3 – Impact of an increase in the demand for quality goodson the number of managers

Impact of greater demand for the ‘quality’ good

• More people become managers

• Wages of managers go up

• Inequality between workers and managers go up

• Inequality between 2 workers or 2 managers is unchanged

q

Ln ω(q)

Figure 7.4: Impact of an increase in the demand for quality goodson the distribution of income.

w

Workers Managers

Impact of greater replicability

• Assume γ and c0 fall such that c’ falls• Given p, each firm in the q sector wants to

produce more• That increases the wages of managers• The supply curve shifts to the left: supply of

managers goes up• Conversely, the same output level can be

produced by fewer firms:– Demand curve shifts to the right– Demand for managers falls

q*

p

Supply of commodities

Demand for commodities

Figure 7.5 – Impact of a greater replicability of quality goods

Demand and Inequality

• If demand is elastic enough:– There are more managers.– Inequality goes up – But all workers gain.

• If demand is inelastic:– Fewer firms/managers.– Wages fall for displaced managers.– Inequality goes up at the top and down at the bottom

• Commodity workers gain as consumers

q

Ln ω(q)

Figure 7.6: Impact of replicability on the distribution of incomeabsent displacement

w

Workers Managers

q

Ln ω(q)

Figure 7.7: Impact of replicability on the distribution of income under displacement

w

Displaced managers

Losers

A

B

Growth and the allocation of talent (MSV, 1991)

• Two activities: Managers vs. Bureaucrats• Managerial reward structure determined by

previous model• Bureaucratic reward structure determined by

other considerations (ex: « fermiers généraux »)• Which occupation the most talented people hold

depends on elasticities• It has important implications for growth• Empirical evidence on lawyers vs. engineers

Hierarchy and span of control

• The model can be extended to hierarchies• The higher one is in the hierarchy, the

more people one controls• Potentially, that increases the rents one

can get• Yields predictions on the links between

earnings, skills, and hierarchical position• Span of control affects the distribution of

income

Intuitively:

A simple approach, adapted from Rosen (1982)

• At each hierarchical level, people produced a good, called productivity

• It is used by people below, affecting their own productivity

• Productivity is sold to the people below at some market price

• But as there is a single person in charge of a level, each productivity level has its own price

The model

• N+1 stages of production• At each stage, worker productivity depends on

his/her skill and supervisor’s productivity• At the last stage, productivity determines output

of the final good• At the top, productivity is a function of skill only• Supervisor’s productivity is non-rival and affects

all workers below• Each supervisor controls n workers• n = span of control

Stages of production:

• Production function for productivity

• Productivity at the top

• Output at the bottom

The market for productivity

• In principle, each firm should decide what kind of worker to allocate at each level and how many levels to have complex assignment problem

• To simplify, we assume a market for each productivity level

• Given equilibrium prices, people endogenously sort themselves into different hierarchical levels

Determination of the wage schedule (I)

• Maximization problem of a worker with skill s

• For production workers:

• For the others:

Determination of the wage schedule (II)

• FOC for optimal choice of my supervisor:

• We can get the marginal wage:

• Substituting:• Get a link between

marginal wage in two consecutive levels

Interpretation:

The multiplier effect

• If I control more people, total MWP for my skill level goes up

• My return to skills goes up• Value of my productivity to lower level workers is

proportional to their own return to skills– Return to skills multiplied by a factor when one moves

up– That factor is higher, the higher n

• That reflects the cumulative effect of my productivity on all the workers below me

Assume the multiplier is > 1

• Sorting by skills into hierarchical levels

• More skilled workers end up in higher levels

• The wage schedule has kinks as one goes up in the hierarchy

q

Ln ω(q)

Level 0 Level 1 Level 2 Level 3

Figure 7.9: The distribution ofwages across hierarchical levels

An increase in span of control

• Each skill interval must be wider• The multiplier goes up inequality between ladders

goes up• The least talented workers in a given ladder are

displaced to a lower hierarchical level• Displacement tends to harm the displaced workers• But they are better managed, and get higher productivity• Only the second effect remains for production workers• Inequality falls at the bottom• Losers are the least talented in their own occupation

q

Ln ω(q) Figure 7.10: increase in the span of control: displacement and wage losses

Displaced workers

Job losers

Conclusion

• The superstars model allows to analyse technical change which affects the span of control of talented workers

• While the change is inegalitarian, it is not so uniformly:– Complementary, untalented workers, gain– Displaced talented workers lose– Inequality between the two falls– Inequality rises at the top and between displaced and

nondisplaced talented workers

XI. Complementarities and Segregation by skills

The motivation

• Workers exert spillovers on each other at the workplace

• For that reason, how workers are matched together matters

• We want to know who works with whom: sorting? Mixing?

• How does tecnology affect the pattern of sorting?

• How does sorting affect the distribution of income

A simple model of complementarities and sorting

• Teams of two workers

• Two skill levels qA

(θ), qB (1-θ).

• Free entry of firms

• 3 potential types of firms

Equilibrium wages

• No firm type can make a strictly positive profit

• Existing firm types make zero profits

Equilibrium in the labor market

• Supply = demand for each type of worker

Segregated equilibria

Complementarity leads to segregation

• Segregated equilibria arise if skills are complements

• The condition holds for any pair iff

• Firms who hire a high skilled in one position are willing to pay more to increase the skill at the other position

Mixed equilibria

Computing wages

• Unless θ = ½, segregated firms must also exist

• So we can solve the model by writing that the wage of the most abundant factor is equal to f(q,q)/2

Equilibrium when θ = ½

Summary:

• The pattern of segregation depends on the cross-derivative of f

• Complementarity segregation• Substitutability mixing• Furthermore, the assignment is efficient

– The segregation condition states that two pairs of workers produce more if matched within types than across types

– Spillovers of workers on each other are entirely internalized by firms

Is segregation worrisome?

An example

• CES production

• X-derivative:

• Segregation condition

η

φ

Figure 8.1: Mixing vs. Segregation

Segregation

Mixing

O

A

Wages:

Storm in a teapot?

• No discontinuity in the wage distribution when one goes through the segregation frontier

• An arbitrary level of inequality can be obtained in the non segregated zone

• A given level of SBTC is not more inegalitarian in the segregated zone– In fact, segregation shelters the unskilled against a

rise in the outside option of the skilled!

Banning segregation?

• Assume we ban BB matches

• If θ > 1/2, the wage of A is unchanged, and that of B falls

• If θ < ½, the wage of B falls to zero

• If we ban AA matches under θ < ½, the A worker lose and inequality goes up

• In all cases we run into the problem that segregation is output-maximizing

The n-worker case

• Production function

• Continuum of skill levels

• Worker assignment measure μ(…)

• Wage schedule ω(q)

What is an equilibrium?

Efficiency

• The equilibrium maximizes total output• Idea: In an alternative assignment, each

firm cannot make positive profits• Hence, total alternative output cannot

exceed total equilibrium wages• But total equilibrium wages = total

equilibrium output• Again, all interactions internalized by firms

efficiency

Constructing a segregated equilibrium

• Wages are determined by

• Condition (1) clearly holds• Condition (3) holds if μ is correctly defined• For (2) to hold, we need

Generalizing the sorting condition

• If all X-derivatives are positive, then the inequality holds and a segregated equilibrium exists

• Idea of the proof:– 1. Show that because of positive XDs, swapping high

productive workers with a firm whose skill distribution dominates increases total output

– 2. Take n firms employing n2 workers, start from n identical firms employing the LHS mix, and perform a sequence of output-increasing swaps to reach the RHS

O-ring

• Batches of output

• Serial processing of tasks

• A failure to process 1 task destroys the whole batch

• Workers with more human capital are less likely to fail

O-ring: a case for segregation

• The production function is the product of skills

• Cross-derivatives are >0

• The outcome is segregated

• Wages are simply given by

Discussion:

• Increasing quality at one task is more valued when other workers have higher quality

• Bottleneck effect: gains from reducing failure are larger if other workers have a low failure rate

• Skill elasticity of wages = number of tasks• The division of labor is intrinsically inegalitarian• « Span of control » effect: if n is larger, failure

destroys more output, return to worker quality greater

Parallel processing

• Assume n workers work to solve a problem• Output is successfully produced if any one of

them finds the solution• Otherwise, output is zero• In such a case, cross-derivatives are negative

– The value of additional problem solving skills is lower if other workers more likely to solve the problem

• Equilibrium is not segregated

The parallel processing model

Intermediate report

• We can extend the model to n workers, if we make assumptions that guarantee segregation

• Let us now take another route and see if we can establish results by restricting the form of interactions between workers

• Assume, for example, that total output only depends on the average quality of the workers

The « average interaction model » (Saint-Paul, 2001)

• By changing the way skills are measured, we encompass a number of special cases:– CES– O-ring

• All skills are complements or substitutes depending on local curvature of f

• We know that there is segregation if f convex throughout

• What happens if f has an arbitrary shape?

• Intuitively, convex zones should lead to segregation, concave ones to mixing

• Can we check that intuition?

q

y

Figure 8.2: The output scheduleO -

f()

Setting up the problem

The first order conditions

• They define a MWP for each type of worker by each type of firm

• In equilibrium MWP = wage if type employed by the firm

• MWP <= wage if type not employed by the firm

• Because of the average interaction structure, firm type unidimensional and defined by its average skill level

The Lagrange multiplier

• It is the shadow price of the (exogenous) size of the firm

• It can be interpreted has a shadow cost of a slot in the firm

• MWP = effect of worker on average quality – shadow cost of a slot

The MWP reflects the pricing of inter-worker spillovers

• An average worker gets the average output• An above average worker improves average

worker quality, and gets a premium• A below average worker reduces average

worker quality, and gets taxes• The price of quality offered by the firm is the

marginal effect on average output on an improvement in worker quality

Equilibrium assignment

• The MWP to pay schedule is tangent to the average output schedule at the firm’s type

• It coincides with the wage schedule if workers employed by the firm

• Otherwise, it must be below the wage schedule

• Wage schedule = upper envelope of the MWP schedules

q

Figure 8.3: The wages offered by a given firm

O

f(q)/n

ω(q)

¯

q̄

Properties of the equilibrium

• The wage schedule is convex• By free entry, it is above the average

output schedule• The allocation of workers to firms involves

clustering by skills– Due to the linearity of each MWP schedule– The cluster where a firm type recruits

corresponds to the linear portion of its MWP schedule where it does hire workers

q

Figure 8.4: The wage schedule is convex

O

Wage paid by firm i

Wage offered by firm i

Average output

Firm 1

Firm 2

Firm 3

Understanding clustering

MWP schedule

ω(q)

ω(q’’)

q q’ q’’

ω(q’’)

Figure 8.5: the wage schedule cannot be convex if the firm isnot hiring from a connected set of workers.

Equilibrium

• An equilibrium is a set of firm types, with a measure for each type, such that– The associated wage schedule is above the

average output schedule– The implied distribution of employment

matches the distribution of available skill levels

Concavity

• At most a single firm type in equilibrium

• Otherwise, the high skill firm would bid more for low skilled than the low skilled firm, and vice-versa

• Firm type = average skill level in the economy

• There is a single cluster (unitary zone)

q

y/n

Figure 8.6: A single unitary zone

O ¯ = E(q)

Wages

Average output

Wages with a less skilled population

The convex case (O-ring)

• The tangent is always below average output schedule

• I can hire at most 1 category of workers• Continuum of clusters and firm types• Each cluster = a single point

(hypersegregation)• Wage schedule = average output schedule• Firms hire a single worker type

q

Figure 8.7: Hypersegregation

O -

Wages = averageoutput

MWP schedule for firm q-

The S-shaped case

• A unitary zone of skilled workers• A hypersegregated zone of unskilled

workers• The average skill level in UZ = firm type in

UZ• By concavity, that requirement yields a

unique equilibrium• In concave hump wide enough,

hypersegregated zone disappears

q

y/n

Figure 8.8: The S-shaped case with two zones

O¯

Wages

Average output

Hypersegregatedzone

Unitary zone

q

y/n

Figure 8.9: The S-shaped case with a single unitary zone

O¯ = E(q)

Wages

Average output

The geometry of segregation

• In general, humps give rise to unitary zones, and troughs give rise to hypersegregated zones

• But a hypersegregated zone between two humps may disappear if trough not too strong

• And two consecutive humps may even share the same cluster (dual cluster)

q

y/n

Figure 8.11: The two humps case, A.

O ¯

Wages

Unitary zone 1

Average output

Unitary zone 2

Firm type 1Firm type 2

Hypersegregated zone

q

y/n

Figure 8.12: The two humps case, B.

O ¯

Wages

Unitary zone 1

Average output

Unitary zone 2

Firm type 1Firm type 2

q

y/n

Figure 8.13: The two humps case, C.

O ¯

Wages

Dual cluster

Average output

Firm type 1Firm type 2

q

y/n

Figure 8.14: The two humps case, D.

O ¯

Wages

Unitary zone

Average output

Back to parallel processing

• Production function

• Change of variable

• New production function

• Concavity single unitary zone

• Wages:

Inequality and the number of workers

• Under serial processing, n increases inequality

• Under parallel processing, n reduces inequality

• Additional problem-solving skills are less valued as more people are working on the problem

Parallel processing with TFP effects

• Workers solve problems and also produce output

• Higher skilled workers have a higher probability of finding a solution

• At the same time, they produce a greater flow of goods

The S-shape

• The formula indicates that average output schedule is S-shaped

• Low levels of skills: problem-solving value of increasing skills does not fall sharply with average skill level overall increasing returns

• The contrary occurs at high skill levels

q

y/n

Figure 8.15: Effect of a deterioration in the skill distribution on wages

O¯

Old wages

Average output

Insulated workersNew wages

Displacedlosers

Non displacedlosers

Gainers

q

y/n

Figure 8.17: Impact of an increase in n on segregation and wages

O¯

Old wage schedule

New wage schedule

Old unitary zone

New unitary zone