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  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 1 st, in my competitive model, low quality producers earn zero profit  stars earn all profit. 2 nd, 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. 3 rd, 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  P  AC. 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 = z  q  + 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 - z  q  - F Find  -max. q & substitute into R to get R*, which yields:

22  Since  > 1, if  > , > 0.  If  < 1: > 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, z 0, 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, z 0, to equal the height of the minimum point of average cost, P 0.

25 Arbitrage  : P(z) = where k (introduced earlier)  P 0 /z 0. 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: P D = f(Q, ), with P D = the demand price for the average quality in the market ( ) & Q = market output.

27 Adjustment to market equilibrium  Suppose z 0 = 1 & P 0 = $10. Minimum quality sells for $10, & higher quality (z) sells for P(z) = $10z. Suppose is initially = 2 & P( ) = $20.

28 q $ AC P 0 = $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’.

30 S D D’ Q* S’ Q P $20 $30 a b c d Figure 1 Market

31  For ex., if P( ) = $24 after entry, since /z 0 = 3, P(z 0 ) = $8 < AC, so  < 0 for those with z = z 0.

32 Thus, the lowest quality sellers exit, market supply decreases, increases, & market demand increases until, at the new level of, P( ) = P 0 /z 0.

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( ) .

34 A Model Let total cost, C, = q 2 + F. AC = q + F/q. Min. pt. of AC: q = F 1/2, so P 0 = 2F 1/2. Let inverse mkt. demand be: P D = [1000 – Q]. In long-run equilibrium, P D = P( ), so solve inverse mkt. demand for Q:

35 Q = A – P( )/ = A – P 0 /z 0, 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).

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 2 nd firm will have a profit-maximizing q that is 4 times that of the 2 nd 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 z 0 = 1 & P 0 = $10, we have: N = max(0, 2[99 – Q star /10]), where Q star = output of all those with z > z 0.

39 Assume Q Star < 990, so some sellers with the lowest quality (z 0 ) exist in long-run equilibrium. Suppose all stars are identical, & consider some examples. Note, given MC, q(z 0 ) = 5, z 0 = 1, &, given mkt. demand, Q = independent of.

40

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 z Star = 9, so q Star = 45).

42 What is required for Superstar effects? With Cost = z  q , in the examples above, I used  = 0 &  = 2. If 0 <  < 1, &  > 2, we would not have  -max. q linear in z, rather  2 q*/  z 2 < 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.