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A Competitive Model of Superstars. Timothy Perri Department of Economics Appalachian State University Presented at Virginia Tech January 21, 2006. Sherwin Rosen ( AER , 1981) developed the notion of superstars. Rosen assumed more talented individuals produce - PowerPoint PPT Presentation
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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
40
# 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%
41
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.