A Competitive Model of Superstars

Timothy Perri

Department of Economics

Appalachian State University

Presented at Virginia Tech

January 21, 2006

2

Sherwin Rosen (AER, 1981) developed the notion of superstars.

Rosen assumed more talented individuals produce

higher quality products.

Superstar effects imply earnings are

convex in quality, the highest quality producers earn a

disproportionately large share of market earnings, & the

possibility of only a few sellers in the market.

3

R* = revenue given the profit-maximizing quantity

z = product quality

R*

4

Rosen argued superstar effects are the result of two

phenomena: imperfect substitution among products,

with demand for higher quality increasing more than

proportionally, and technology such that one or a few

sellers could profitably satisfy market demand.

5

Herein, a competitive model is developed in which:

1) there are many potential and active firms;

2) some fraction of the potential producers with the lowest quality level could satisfy market demand;

3) complete arbitrage occurs between prices of goods with different quality; and

4) a few firms with higher quality earn a disproportionately large share of market revenue because their revenue increases with quality at an increasing rate.

6

The usual explanations for superstar

effects---imperfect substitution between

sellers, and some form of joint

consumption, with marginal cost

declining as quality increases---are not

necessary.

7

A firm’s revenue can be positive and

convex in quality when cost increases

in quality at a decreasing rate.

Without the requirements of imperfect

substitution and joint consumption,

there may be many markets that could

contain superstar effects.

8

Evidence

Krueger (JOLE, 2005) identifies significant

superstar effects for music concerts that have

become even larger in recent years. He argues

the time and effort to perform a song should

not have changed much in over time.

9

It is also unlikely the cost of performing a

song depends significantly on the quality

of the musicians. Further, the technology

of reaching more buyers for a live performance

is much different than it is for selling additional CDs .

10

“Pavarotti can, with the same effort, produce

one CD of Tosca or 100 million CDs of Tosca.

...if most view Pavarotti as {even slightly}

better {than Domingo}, he will sell many more

CDs than Domingo and his earnings will be

many times higher...”

(Lazear, p. 188, 2003)

11

Krueger (2005) reports revenue for music concerts from 1982 to 2003. In 1982, the top 5% (in terms of revenue) of artists

earned 62% of concert revenue. For 2003,

the corresponding figure was 84%.

12

An example

In Rosen (1981), imperfect substitution

between quality levels would produce star

surgeons. However, if star surgeons have quality

levels significantly higher than non-star surgeons,

then imperfect substitution is not necessary for stars

to have significantly higher revenue than non-stars.

13

The term “superstar” will be used when

revenue increases & is convex in quality, &

a few sellers earning a large % of market revenue.

Rosen used profit (), but revenue (R) is used herein.

WHY?

14

1st, in my competitive model, low quality

producers earn zero profit stars earn all profit.

2nd, in the special case in Rosen closest to the

model herein, revenue and profit are identically

affected by quality, as is true in my competitive model.

3rd, earnings reported for top performers in entertainment

and sports are not net of cost. The data on concert

earnings from Krueger (2005) involve revenue.

15

Rosen (1981) argued his model involved competition.

However, different quality levels were imperfect

substitutes (with the larger the difference in quality the

worse subs. goods were), & the threat of POTENTIAL

ENTRY disciplined existing producers.

16

Adler (2005) argued there

would not be relatively high earnings

for superstars unless there are

significant quality differences

between sellers.

17

With several sellers of similar quality,

if MC declines with firm output,

competition PAC. One “superstar”

may survive and sell most of market Q,

but it will not have > 0.

18

However, if quality levels are

not similar between firms,

there is no competition in

Rosen’s model.

19

Cost and superstar effects

Let C = a firm’s total cost, q = output, z = quality, & F = fixed cost:

C = zq + F,

where > 1 & could be positive, negative, or 0.

20

A firm’s price is P(z) = kz, with k (> 0)

a positive constant to be determined later.

Rosen (1981) argued superstar effects

occur in a market when “...fewer are

needed to serve it the more capable

they are.” This means marginal cost is

inversely related to quality, or < 0.

21

= kzq - zq - F

Find -max. q & substitute into R to get R*, which yields:

1

1

11

1

zk1

1

z

*R

1

2

12

1

1

2

2

zk1

11

z

*R

22

Since > 1, if > ,

z

*R

> 0.

If < 1:

2

2

z

*R

> 0.

Thus, cost could increase in Z (at a

decreasing rate) & still have R* positively

& convexly related to Z.

23

Market equilibrium

Suppose most sellers (non-stars)

have the minimum quality level, z0, and a

few sellers (stars) have higher quality.

Free entry and exit of non-stars occurs.

Assume cost is independent of quality,

which is not necessary for the existence

of superstar effects.

24

Each firm has a U-shaped AC

curve. Entry or exit of non-stars will

force the long-run price of the lowest

quality level, z0, to equal the height of

the minimum point of average cost, P0.

25

Arbitrage :

P(z) = 0

0

Pz

z

where k (introduced earlier) P0/z0.

Arbitrage determines relative Ps, &

free entry/exit of non-stars determines

absolute Ps.

26

Market demand depends on the average quality sold, with inverse market demand:

PD = f(Q, ),

with PD = the demand price for the

average quality in the market ( )

& Q = market output.

z

z

27

Adjustment to market equilibrium

Suppose z0 = 1 & P0 = $10. Minimum

quality sells for $10, & higher quality

(z) sells for P(z) = $10z. Suppose

is initially = 2 & P( ) = $20.

z

z

28

q

$AC

P0 = $10

firm

29

If entry raises to 3,

even if the elasticity of

demand with respect to = 1

(a to b in Figure 1), P( ) will

rise to < $30 because:

1) supply is not vertical, &

2) supply increased to S’.

z

z

z

30

S

D

D’

Q*

S’

Q

P

$20

$30

a

bc

d

Figure 1

Market

31

For ex., if P( ) = $24 after entry,

since /z0 = 3, P(z0) = $8 < AC,

so < 0 for those with z = z0.

z

z

32

Thus, the lowest quality sellers

exit, market supply

decreases, increases, &

market demand increases until,

at the new level of ,

P( ) = P0/z0.

z

z

z z

33

Only if the elasticity of market demand

with respect to z is equal to x (x > 1) would

price as much as . If this elasticity > x,

then P( ) would faster than , low

quality sellers would have > 0, entry

would occur at this quality level, market

demand would , & P( ) .

zz

z

z

34

A Model

Let total cost , C, = q2 + F. AC = q + F/q.

Min. pt. of AC: q = F1/2, so P0 = 2F1/2.

Let inverse mkt. demand be:

PD = [1000 – Q].

In long-run equilibrium, PD = P( ), so solve

inverse mkt. demand for Q:

z

z

35

Q = A – P( )/ = A – P0/z0,

due to arbitrage. The above Q is

the long-run equilibrium point on

mkt. demand: where the market

clears, = 0 for non-stars, &

arbitrage determines P(z).

z z

36

The total # of firms in long-run

equilibrium depends on the distribution

of stars. The # of non-stars, N, adjusts

to maintain zero for non-stars.

37

Given the assumed cost equation,

MC is independent of z, &, since P(z)

is linear in z, a firm with, say, 4 times

the quality of a 2nd firm will have a

profit-maximizing q that is 4 times

that of the 2nd firm.

38

Long run supply comes from adding

each firms MC (depending on the long-

run equilibrium # of firms). Setting

supply & demand = determines N. With

z0 = 1 & P0 = $10, we have:

N = max(0, 2[99 – Qstar/10]),

where Qstar = output of all those with z > z0.

39

Assume QStar < 990, so some sellers with the

lowest quality (z0) exist in long-run equilibrium.

Suppose all stars are identical, & consider some

examples. Note, given MC, q(z0) = 5, z0 = 1, &,

given mkt. demand, Q = 990---independent of .z

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# of stars

zstars Qstars QNon-stars # of Non-stars

Stars’ % of total # of firms

Stars’ % of Q Stars’ % of R

8 5 200 790 158 5% 20% 56% 8 6 240 750 150 5% 24% 66% 8 9 360 630 126 6% 36% 84%

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NOTE: in the examples considered in the table,

no one firm sells as much as 5% of the total

amount sold (the case when zStar = 9, so

qStar = 45).

42

What is required for Superstar effects?

With Cost = zq, in the examples above,

I used = 0 & = 2. If 0 < < 1, &

> 2, we would not have -max. q linear

in z, rather 2q*/ z2 < 0.

43

Superstar effects still will exist (but will be

smaller) if:

1) significant quality differences exist between sellers;

2) the elasticity of total cost with respect to quality is less than 1; &

3) total cost does not increase too rapidly as output increases.