X. The economics of superstars

1. X. The economics of superstars

2. 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

3. 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

4. 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

5. 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

6. Determination of the wage schedule

7. 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

8. 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

9. A specification • Lower γ greater replicability  more convex and inegalitarian • wage schedule

10. ω q Figure 7.1: Impact of a reduction in γon the distribution of income

11. 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

12. A counter-example • The increase in size induced by a fall in φin itself increases the local elasticity of costs

13. 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

14. Ln ω(q) w Workers Managers q Figure 7.2: Income distribution and the allocation of talent

15. 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

16. p Demand for commodities Supply of commodities q* Figure 7.3 – Impact of an increase in the demand for quality goods on the number of managers

17. 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

18. Figure 7.4: Impact of an increase in the demand for quality goods on the distribution of income. Ln ω(q) w Workers Managers q

19. Impact of greater replicability • Assume γand 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

20. p Demand for commodities Supply of commodities q* Figure 7.5 – Impact of a greater replicability of quality goods

21. 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

22. Figure 7.6: Impact of replicability on the distribution of income absent displacement Ln ω(q) w Workers Managers q

23. Figure 7.7: Impact of replicability on the distribution of income under displacement Ln ω(q) B A w Displaced managers Losers q

24. 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

25. 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

26. Intuitively:

27. 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

28. 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

29. Stages of production: • Production function for productivity • Productivity at the top • Output at the bottom

30. 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

31. Determination of the wage schedule (I) • Maximization problem of a worker with skill s • For production workers: • For the others:

32. 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

33. Interpretation:

34. 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

35. 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

36. Ln ω(q) Figure 7.9: The distribution of wages across hierarchical levels q Level 1 Level 2 Level 3 Level 0

37. 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

38. Ln ω(q) Figure 7.10: increase in the span of control: displacement and wage losses Job losers q Displaced workers

39. 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

40. XI. Complementarities and Segregation by skills

41. 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

42. 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

43. Equilibrium wages • No firm type can make a strictly positive profit • Existing firm types make zero profits

44. Equilibrium in the labor market • Supply = demand for each type of worker

45. Segregated equilibria

46. 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

47. Mixed equilibria

48. 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

49. Equilibrium when θ = ½

50. 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