The Economics of Superstars
Sherwin Rosen
The American Economic Review, Vol. 71, No. 5. (Dec., 1981), pp. 845-858.
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http://www.jstor.org Mon Dec 3 22:55:54 2007 The Economics of Superstars
The phenomenon of Superstars, wherein instrument is also on the order of only a few relatively small numbers of people earn enor- hundred (and much smaller for instruments mous amounts of money and dominate the other than voice, violin, and piano). Per- activities in which they engage, seems to be formers of first rank comprise a limited increasingly important in the modern world. handful out of these small totals and have While some may argue that it is all an illu- very large incomes. There are also known to sion of world inflation, its currency may be be substantial differences in income between signaling a deeper issue.' Realizing that world them and those in the second rank, even inflation may command the title, if not the though most consumers would have diffi- content of this paper, quickly to the scrap culty detecting more than minor differences heap, I have found no better term to describe in a "blind" hearing. the phenomenon. In certain hnds of eco- (iii) Switching to more familiar territory, nomic activity there is concentration of out- sales of elementary textbooks in economics put among a few individuals, marked skew- are concentrated on a group of best sellers, ness in the associated distributions of income though there exists a large number of very and very large rewards at the top. good and highly substitutable alternatives Confidentiality laws and other difficulties in the market (the apparent inexhaustable make it virtually impossible to obtain sys- supply of authors willing to gamble on tematic data in this field. However, consider breaking into the select group is one of the the following: reasons why so many are available). A small (i) Informed opinion places the number number of graduate schools account for a of full-time comedians in the United States large fraction of Ph.D.s. A relatively small at approximately two hundred. This is per- number of researchers account for a large haps a smaller number than were employed fraction of citations and perhaps even articles in vaudevillian days, though it hardly can be written. maintained that the demand for (intended) Countless other examples from the worlds comic relief is in a state of secular decline. of sports, arts and letters, and show business Some of the more popular performers today will be well known to readers. Still others earn extraordinary sums, particularly those can be found in several of the professions. appearing on television. The capacity for There are two common elements in all of television to produce high incomes is also them: first, a close connection between per- manifest in the enormous salaries paid to sonal reward and the size of one's own network news broadcasters. market; and second, a strong tendency for (ii) The market for classical music has both market size and reward to be skewed never been larger than it is now, yet the toward the most talented people in the activ- number of full-time soloists on any given ity. True, standard theory suggests that those who sell more generally earn more. But that *University of Chicago and NORC. I am indebted to principle applies as well to shoemakers as the National Science Foundation for financial support, to rock musicians, so something more is and to Gary Becker, David Friedman, Robert J. Gordon, Michael Mussa, Edward Prescott, and George Stigler for involved. In fact the competitive model is helpful discussion and comments. virtually silent about any special role played 'That escalation is not confined to wars and prices is by either the size of the total market or the established by the fact that Stars would have sufficed amount of it controlled by any single person, not long ago. Academics have a certain fondness for because products are assumed to be undif- Giants, while businessmen prefer Kings. Obviously there is a fair bit of substitution among all these terms in ferentiated and one seller's products are as- depicting related data in different contexts. sumed to be as good as those of any other. 846 THE AMERICAN ECONOMIC RE VIEW DECEMBER 1981
The elusive quality of "box office appeal," quences of Superstars. Since R(q) is the the ability to attract an audience and gener- transformation that takes the distribution of ate a large volume of transactions, is the talent to the distribution of rewards, convex- issue that must be confronted. Recognition ity implies that the income distribution is that one's personal market scale is important stretched out in its right-hand tail compared in the theory of income distribution has a to the distribution of talent. Hence a genuine long history, but the idea has not been devel- behavioral economic explanation is provided oped very extensively in the literat~re.~I for differential skew between the distribu- hope to fill in some of the gaps in what tions of income and talent, a problem that follows. has been an interesting and important preoc- The analytical framework used is a special cupation of the literature on income distribu- type of assignment problem, the marriage of tion down through the years.3 Convexity of buyers to sellers, including the assignment of R(q) literally means that small differences in audiences to performers, of students to talent become magnified in larger earnings textbooks, to doctors, and so forth. differences, with great magnification if the Rest assured that prospective impresarios will earnings-talent gradient increases sharply receive no guidance here on what makes for near the top of the scale. This magnification box office appeal, sometimes said to involve effect is characteristic of the phenomenon a combination of talent and charisma in under consideration. uncertain proportions. In the formal model Convexity of returns and the extra skew it all that is taken for granted and represented imparts to the distribution of earnings can be by a single factor rather than by two, an sustained by imperfect substitution among index q labeled talent or quality. The distri- different sellers, which is one of the hall- bution of talent is assumed to be fixed in the marks of the types of activities where Super- population of potential sellers and costlessly stars are encountered. Lesser talent often is a observable to all economic agents. Let p be poor substitute for greater talent. The worse the price of a unit of service (for example, a it is the larger the sustainable rent accruing performance, a record, a visit, etc.) and let m to higher quality sellers because demand be the size of the market, the number of for the better sellers increases more than "tickets" sold by a given seller. Then an proportionately: hearing a succession of overall market equilibrium is a pair of func- mediocre singers does not add up to a single tions p(q) and m(q) indicating price and outstanding performance. If a surgeon is 10 market size of sellers of every observable percent more successful in saving lives than talent and a domain of q such that: (a) all his fellows, most people would be willing to sellers maximize profit and cannot earn larger pay more than a 10 percent premium for his amounts in other activities, and (b) all buyers services. A company involved in a $30 mil- maximize utility and cannot improve them- lion law suit is rash to scrimp on the legal selves by purchasing from another seller. talent it engages. Properties of sellers' maximum net reve- Imperfect substitution alone implies con- nue functions, R(q), will have special inter- vexity and provides a very general explana- est. Specifically, convexity of this function tion of skewed earnings distributions whlch describes much of the observable conse- applies to myriad economic service activities.
*~lbertRees is a good introduction to the size distri- 'Few economic behavioral models exist in the liter- bution of income. The selectivity effects of differential ature. On this see Harold Lydall. Jacob Mincer has talent and comparative advantage on the skew in in- shown that investment can produce skewness through come distributions are spelled out in my 1978 article, the force of discounting, and established that as an also see the references there. Melvin Reder's survey important source of skewness empirically. Learning is touches some of the issues raised here. Of course social not treated here because those issues are well under- scientists and statisticians have had a long standing stood, whereas the assignment problem has received fascination with rank-size relationships, as perusal of the little attention. Some recent works, but with different many entries in the Encyclopedia of the Social Sciences focus and emphasis than is discussed here, are Gary will attest. Becker (1973), David Grubb, and Michael Sattinger. VOL. 71 NO. 5 ROSEN: ECONOMICS OF SUPERSTARS 847
However, preferences alone are incapable of I. Structure of Demand explaining the other aspect of the Superstar phenomenon, the marked concentration of Imperfect substitution among quality dif- output on those few sellers who have the ferentiated goods in the same product class most talent. Thls second feature is best arises from indivisibilities in the technology explained by technology rather than by of consumption. No satisfactory analytical taste^.^ In many instances rendering the specification exists in the literature, because service is described as a form of joint con- indivisibilities lead to nonadditivities in pref- sumption, not dissimilar to a public good. erence relations whlch are analytically intrac- Thus a performer or an author must put out table.5 Yet some specific model is required to more or less the same effort whether 10 or make any progress on this problem. My solu- 1,000 people show up in the audience or buy tion to thls dilemma is to adopt a smooth the book. More generally, the costs of pro- quantity-quality substitution technology and duction (writing, performing, etc.) do not introduce the indivisibility through a fixed rise in proportion to the size of a seller's cost of consumption per unit of quantity. market. Consumers' attempts to minimize consump- The key difference between this technol- tion costs gives an extra competitive ad- ogy and public goods is that property rights vantage to higher quality sellers. However, it are legally assigned to the seller: there are no is a surprising implication of the analysis issues of free riding due to nonexclusion; that this form of indivisibility is not crucial customers are excluded if they are unwilling to the central conclusions, so that true non- to pay the appropriate admission fee. The additivities would only strengthen the argu- implied scale economy of joint consumption ment. allows relatively few sellers to service the Assume the consumer has a well-behaved entire market. And fewer are needed to serve weakly separable utility function u = it the more capable they are. When the joint u(x, g(n, z)), where x is a composite com- consumption technology and imperfect sub- modity and y =g(n, z) has the natural inter- stitution features of preferences are com- pretation of consumption of "services" of the bined, the possibility for talented persons to type in question. n is the quantity purchased, command both very large markets and very a measure of exposure to a seller, such as a large incomes is apparent. patient visit, a performance, etc.; and z is the A theory of the assignment of buyers to quality of each unit of exposure. Quantity- sellers is required to make these ideas pre- quality substitution requires that g(.) is in- cise. The demand and supply structure of creasing in both of its arguments and that one such model is set forth in Sections I and a2g/ana~>o. 11. The nature of market equilibrium and its This specification has the virtue of being implications for income and output distribu- simple, at the cost of ignoring some details tions are discussed in Sections I11 and IV. and not being perfectly general. The defini- Comparative static predictions of the model tions of markets are left somewhat vague: for are sketched in Section V and conclusions example, for some purposes it is sufficient to appear in Section VI. think about the market for novels as a whole and for others distinguishing between mys- teries, romances, and so forth is necessary. 4Milton Friedman proposed a model based on pref- hi^ is simply treated by allowing to be a erences for risk taking, but did not explain why or how the market sustains the equilibrium ex post with few vector and is therefore ignored. There are sellers earning enormous incomes (for example, why the several dimensions to quantity in any specific losers in the lottery rest content with such low incomes application which might be treated in a simi- if they have the same talents as the winners). Issues of uncertainty that make these elements of supply more interesting are abstracted from here. A model of prizes 'some of the thorny issues of primitives in problems based on effort-incentive monitoring and the principal of product differentiation are discussed from the point agency relation is found in my article with of view of the theory of measurement by Manuel Edward Lazear. Trajtenberg. 848 THE AMERICAN ECONOMIC REVIEW DECEMBER 1981 lar manner. For example, most people do not and the wage rate is w,then s= tw. Measur- purchase more than one copy of an author's ing prices in units of x, the budget constraint book but may buy several different books is written by the same author. Or there may be (1) I=x+ (p+s)n preferences for variety. But these considera- tions are less than compelling in markets for professional services where direct personal contact between buyers and sellers is re- where I is full income and u is the full price quired. Preferences for variety per se cannot of services directly implied by the multiplica- be treated in a quantity-quality substitution tive specification y=nz (herein lies the ana- model, and since the generalization of in- lytical value of that assumption). creasing the dimensionality of exposure is Marginal conditions for consumer choice clear enough in any given case, it is ignored are too. It is doubtful whether the general nature (2) u,/ux=dp(z)/dzforz; of the results are greatly affected by these simplifications. u, /ux =(p+s)/z for n. A cardinal measure of quality or talent must rely on measurement of actual out- Combining these two, choice of z solves comes. Taken to extreme, this view would define the talent distribution as the realized output or income distribution. However, that goes too far because it ignores the fact that Choice of n follows from the requirements more talented people typically command that the marginal rate of substitution be- greater cooperating resources in producing tween y and x equals the relative marginal observed outcomes and it refers to all con- cost v=( p +s)z. The schedule p(z) is the sumers as a group rather than to any one of same for all buyers. It maps the talent of a them. The service flow y is a natural personal seller into the unit price charged for that outcome measure in this case and is the quality of service. Therefore optimal choice prime candidate for scaling talent, so long as of z in (3) depends only on s and not on the n is held constant in the imputation to obtain form of the utility function under the separa- the right ceteris paribus conditions. Still, the bility assumption. Condition (3) balances measure is strongly dependent on n unless larger direct costs of greater talent against g(n, z) is multiplicatively separable. To avoid larger indirect costs of greater quantity and ambiguity I restrict g(n,z) to the form zf(n), lesser talent. For example, customers with so that relative talent is defined indepen- larger s prefer more talented sellers to econo- dently of n (since y is the product of a mize on consumption time in this specifica- function of n and another function of z, tion. Finally, all effects of intensity of prefer- talent can always be rescaled to be the func- ences are absorbed in choice of the quantity tion of z itself, for example, if y=f1(zf)f,(n), consumed, given the optimum value of z change the scaling of z' by defining zrfl(z')). determined by condition (3). The properties of f(n) play no important Because it plays an important role in the role in this analysis, so it is assumed to be analysis below, suppose the equality in (3) linear. Therefore y =nz, which is the familiar held for all possible values of z, not just for efficiency units specification. This is a very one of them. Evidently that occurs only if strong form of substitution which obviously p(z) happens to follow a definite functional works in the direction of spreading sales form; the one satisfying (3) interpreted as a around all qualities of sellers, not concentrat- differential equation for all z. Integrating ing them among the top, and is a weak and simplifying equation (3) yields specification in that sense. The cost of one unit of service of a given quality is its price p(z) plus a fixed cost s. For example, if each unit requires t hours The full price u is the constant of integration, VOL. 71 NO. 5 ROSEN: ECONOMICS OF SUPERSTARS 849 since u=(p+s)/z by definition. If market mies reflect a type of degradation of services prices line up as in (4), the consumer is a seller supplies to each of his customers as indifferent among all values of z that appear the number of contacts expands. There are on the market, since (3) is an identity. There- two fundamental reasons for this: fore (4) must be a price-talent indifference First, in cases where duplication is pos- curve, an equalizing difference function, sible, market expansion ultimately requires showing the maximum amount the customer using inferior techniques to render the is willing to pay for alternative values of z at service. It is preferable to hear concerts in a a gven utility index. The larger is v, the hall of moderate size rather than in Yankee smaller the utility index. Finally, if (4) does Stadium. Recordings are a superior way of in fact hold true in the market, too, then reaching a large audience, but are inferior in both equations in (2) reduce to u,/u,=u, so quality to live performances with smaller that y is uniquely determined for the con- audiences. Furthermore, many of the activi- sumer even though z and n are not. ties in question involve certain creative ele- ments so the ultimate negative impact of 11. Structure of Supply: External and market sizes sometimes can be interpreted as Internal Diseconomies the effect of overexposure and repetition. Second, the analysis should not be con- The economic activities under considera- strained to only those activities where some tion invariably involve direct contact of form of cloning is possible. The general model buyers with the seller in one way or another. also applies to cases of one-on-one buyer- If a competitive market was ever impersonal, seller contact, as is true of professional this surely is not it. The seller's choice of services. Here the negative effects of personal market size (volume of transactions) amounts market scale are caused by limitations on the to determining the number of contacts to seller's time. As a doctor's patient load in- make with buyers. In many cases the tech- creases, the amount of direct contact time nology admits a certain lund of duplication available to any person decreases, waiting in which the seller delivers services to many time between appointments and in the office buyers simultaneously, as a form of joint increases, and so forth. Nevertheless patients consumption. Once the author tells his tale may be willing to trade off service time to the publisher, it can be duplicated in against quality of service per unit time. writing as many times as desired. A per- In both cases the quality of service z that former appearing on television literally clones appears in consumers' preferences is itself his performance to whomever happens to produced by both the quality and size of the tune in. The services rendered by any seller market of the seller with whom transactions become more like a kind of public good the occur: z=h(q,m), where q is an index of more nearly the technology allows perfect seller talent or quality and m =m(q) is the duplication at constant cost. total number of units sold by a seller of type Just as it is difficult to find practical ex- q. The arguments above imply az/aq= h, >0 amples of pure public goods in public fi- and az/am=h, GO. Furthermore, I assume nance, so too it is difficult to find them here. that h,, 20: superior talent stands out and In fact services of this type are analogous to does not deteriorate so rapidly with market local public goods, due to ultimate limita- size as inferior talent does. The importance tions on joint consumption economies. To of this assumption will emerge later on. the extent that the technology is subject to Preferences are structured on service flows, congestion, that is, to external diseconomies which in turn depend upon q and m. There- of scale, the required analytical apparatus is fore p =p(q, m) is the unit price charged by a the theory of clubs rather than the theory of seller of quality q selling m units. Competi- pure public goods.6 These external disecono- original reference in the theory of public finance is 6~hata doctor's patients or a performer's fans might James Buchanan. Eitan Berglas and Berglas and David be considered as a club has intuitive plausibility. The Pines, present elegant developments of that model. 850 THE AMERICAN ECONOMIC REVIEW DECEMBER 1981 tion in the market for services implies that In context a more elaborate return specifi- the function p(q, m) is taken as given by a cation decomposes the internal margin above seller. This market is competitive even though into two additional components, one being a seller affects the unit price charged by the size of each act of joint consumption, m,, choosing m. The reason for competition in and the other being the number of such acts, markets of this type is that each seller is m,. In that case the revenue function is closely constrained by other sellers offering similar services. Though sellers of different quality are imperfectly substitutable with each other, the extent of substitution de- creases with distance. In the limit very close where m, p -C, are the "gate" receipts for neighbors are virtually perfect substitutes. each event and C,(m,) is the cost of increas- Assume there is a regular distribution of ing the number of events. This avoids some talent in the population @(q)dq.Then poten- of the dimensionality or units ambiguities in tial substitution is generated by both the (5), as noted in Section I. If all external density @ in the neighborhood dq of q and by diseconomies reside in m, alone and not m, degradation through larger market size of (so that p =p(q, m ,)), then (8) and (5) have better quality sellers some distance above q, very similar implications; only the disecon- and the opposite for those some distance omy associated with each event is somewhat below q. overcome by expanding their number in for- In addition to market size effects on de- mulation (8). This carries over to a case mand, the other factor influencing the out- where the external diseconomy of m, is small. put decision is direct cost of production. Let Otherwise, precise results depend on whether C(m) be out of pocket costs of producing m the effect on performance services of m, are units, with C'aO and C"3O. There are non- stronger than those of m, and on their inter- decreasing (marginal) costs of production- action. It is simplest to merely think of m in internal diseconomies-for the usual rea- (5) as the product of m, and m,, in those sons, including the fact that here the seller cases where this type of decomposition is must work harder as m increases. Assume applicable. also that all sellers have opportunity cost K of working in this sector compared with the 111. Market Equilibrium next best alternative, with K independent of q. A complete closed market solution is A seller of type q chooses m(q) to maxi- available if all buyers have the same fixed mize net revenue cost s, though possibly different marginal rates of substitution between y and x. In that case it is possible to aggregate total services in a single market, with a unique implicit Therefore m(q) is chosen to satisfy market price v which contains all the rele- vant information and acts as a "sufficient statistic." The unit pricep charged by a seller of type q is then constrained to follow (4) so long as independent of market supply conditions. Though n and z are not uniquely determined for any consumer, each one has a regular demand function for services y which de- and R exceeds K. Equation (6) determines pends only upon v. These demands in turn the intensive margin. If R(q) is monotone in can be summed across consumers to obtain q, then free entry determines an extensive the total market demand for services Zy= yd margin as well; the value of q, denoted q,, =F(v). Since consumers are indifferent be- which satisfies both R(q,)= K and (7) simul- tween n and z, the composition of services t aneously. between qualities and quantities are de- VOL. 71 NO. 5 ROSEN: ECONOMICS OF SUPERSTARS 851 termined completely by sellers, who maxi- supplied to the market by all active sellers is mize profit according to condition (6). Indi- vidual supply choices may be aggregated too, this time by integrating the optimum value of zm, the total services a seller supplies to the market, over those values of q which are where q,(v) is the extensive margin. Dif- actually found in the market, weighted by ferentiate with respect to v: the number of sellers of type q, +(q). This sum represents total services supplied to the market, Y"=G(v). The intersection of supply and demand determines v itself. Given this equilibrium, the internal cross-section price, output, and income distributional structure may be examined in detail. where m' =m(q,; v). Condition (9) implies To find the supply decision of each seller that 1 + (m/h)h, is positive. Therefore the at a given value of v, substitute z=h(q,m) second (integral) term in dY/dv is positive. into the equalizing difference function (4). The fact that R is increasing in both q and v Applying (6) and (7) yields from (12) implies that dq,/du A. Internal Diseconomies The cross-section structure of the market equilibrium is most easily established in the Market size increases with q if hqm>O. Next case where there are no effects of a seller's differentiate net revenue R(q) in (5) with market size on service quality.' In that case respect to q, at its maximized value. By the m is not an argument of h(.) and talent is envelope property scaled so that z= h(q)=q. Now the model has a Ricardian flavor, with differential rent sustained by talent induced product differ- entiation. Net revenue is monotonically increasing in Since z rq the unit price charged by sellers talent, since hq>O. Finally, differentiate (12) of talent q is increasing linearly in q at rate v, with respect to q and simplify to obtain from (4); and since price is higher for the better sellers and cost conditions no less favorable, more talented sellers produce more and have larger markek8 Application of (1 1) h his version of the model has a strong family re- where am/aq is defined by equation (1 1). So semblance to a class of problems previously considered in my 1974 article. long as hqq is not sufficiently negative, re- '~hrou~houtthis paper I make the usual club theory ward is convex in q. assumptions and ignore indivisibilities requiring an in- The market supply of services is easily teger number of sellers. This can be problematic when calculated. Let m(q; v) be the solution to (9). the number of sellers is very small, and raises well-known Then the amount of service supplied to the problems in industrial organization about which I have nothng to contribute. The magnitude of the rent of the market by a seller of quality q is lowest rent seller (extensive margin) is the issue. That h(q,m(q;v))m(q;v) and the total amount must be sufficiently small for this analysis to apply. 852 THE AMERICAN EC 'ONOMIC REVIEW DECEMBER 1981 to this case yields am/aq=v/C">O. From (12), R'(q)=vm>O, and R"(~)=V~/C">O, from (13). Not only does rent reward in- crease in talent, but marginal rent reward increases in talent as well. R(q) is convex because both price and quantity increase in q. To see the powerful force of convexity in producing skewness, consider an example where s=O and C(m) is quadratic. Then mavq and both price and quantity increase linearly in q. Therefore, revenue increases in the square of q. In fact ~av~q~/2.A person who is twice as talented as another earns four times more money in this e~ample.~ This case is important in showing that the tendency toward skewed rewards arising from convexity of revenues holds under very gen- eral circumstances: individuals who, by virtue of superior talent and ability in an activity, can sell their services for higher prices have strong incentives to produce more so long as costs are not perfectly correlated with talent. is the most talented of all potential sellers. The increase in both price and quantity with Even though there is one seller, essentially quality implies that talent has a multiplica- competitive market conditions are main- tive effect on reward. It is surprising that the tained by threats of potential entry. tendency toward skewed rewards is not nec- Let N=N(p,q) denote the total market essarily dependent on indivisibilities and oc- demand for quantity at pricep and talent q. curs in the linear efficiency-units case, per- If there were several potential sellers of the haps the weakest possible specification. same talent, only one of them is required to However, no relative skew is implied in the provide the service efficiently, so m -N. This distribution of output in thls case because is seen in Figure 1. Free entry implies that there are no interactive effects in that dimen- total revenue pN must be driven down to sion of the problem. opportunity cost K in equilibrium. This equation, pN= K, is the rectangular hyper- B. Pure Joint Consumption bola in Figure 1. It is competitive supply price. Market equilibrium occurs where de- The effect of scale economy on seller con- mand intersects supply from above. Suppose centration is strikingly seen in the extreme the seller were to charge price p,. Then the case when internal and external disecono- value of sales exceeds K and rents are non- mies vanish, when C(m)=O (nonzero con- zero. Therefore another seller would enter stant marginal costs will do also) and h,=O, and charge a slightly lower price, attracting so z =h(q) =q. Then there literally is public all business away from the initial seller. By goods technology and a single seller services continuation, price must be driven down in the total market in equilibrium. That person equilibrium to p*, rents are driven to zero, there is one seller and potential entry main- he two functions m(q) and R(q) are the trans- tains that situation indefinitely. forms from the distribution of ability to the distribution What happens when sellers have different of output and reward. Inverting and computing the talents? The demand function facing a more Jacobians, the distribution of output is (I/v)$(m/v), able seller is different from the one in Figure the same form as the distribution of talent because m(q) is linear. The distribution of rent is 1 because q is an argument of demand, (V(SR)'/~)+((~R/v)'/'), which is skewed to the right N( p, q). Whether N, is positive or negative, relative to +. less talented sellers are driven out of the VOL. 71 NO. 5 ROSEN: ECONOMICS OF SUPERSTARS market. To see thls, note that R(q)=p. N(p, q) in this case. Therefore Given the structure of demand above, equa- tion (4) implies that ap/aq =v. Furthermore, it is easy to show that the price and quality elasticities of demand for quantities are re- lated to the full price elasticity of services as follows: where 8=p/(p Ss) is the share of full price accounted for by nonfixed costs and Y=Zy with the sum extending over individual con- sumers. The quality elasticity of demand for choose not to compete. q* charges pricep* = quantity is negative if the full price elasticity vq* -s (see equation (4), whereas q would of demand for services is inelastic. Sub- chargep= vq- s. The price differentialp* -p stituting these relations into (14) and sim- =VE is the unit rent accruing to q*. This is a plifying yields small number if e is small. Yet the total rent received by q* is Nve, which can be very large if N is large. Though unit rent is limited by the equalizing difference (4) and the Consider the following two cases: supply (distance) of close competitors, scale (i) Assume +(q) is dense on the interval economies can make total rent very large in [qO,q], where qo is the least talented and q eq~ilibrium.'~ the most talented potential seller. Equation (15) shows that R is increasing in both q and C. External Diseconomies v. For a given value of v all sellers for whom R(q)- K>O would choose to enter and, since External diseconomies support a nonde- Rr>O, they must be selected from the upper generate equilibrium distribution of sellers. tail of +(q). But in equilibrium there is only The spatial structure of the market is il- one seller. Therefore v must adjust so that lustrated in Figure 2. Given the market full R(q)- K=O and all people for whom q To examine the pure effect of externalities into (13)to obtain assume no internal diseconomies, C(m)-0. Two families of curves are shown in Figure 2, one corresponding to the objective func- tion, and the other to the constraint at alter- native values of q. A seller of given talent q, is constrained by both consumer preferences Since the first term in (18)is positive, R(q)is and sellers of other talents to charge prices convex so long as hqq is not sufficiently along the curve- marked v h(q,,m)-s; seller negative. In fact, given the caveat about 9, is constrained by the presence of q, and hqq,RU(q)>O independent of the sign of others to operate along vh(q,m) -s, etc. The h .When hqm should the equilibrium be described when s is distributed in the population of customers? That analysis is more complex because there is no longer a single equilibrium market price for services, v, that summarizes all the infor- mation. Nevertheless, differences in s imply restrictions on market outcomes that actually strengthen the qualitative results. I do not attempt a full analysis here, but the reason is that the market assignment of customers to sellers may force the relationship between p and z to be convex. Therefore the more talented sellers receive even greater rents and service even larger markets than when p is linear in z as in (4). That p(z) must be convex can be sketched as follows: Figure 3 shows the equalizing difference function (4) for two types of V. Comparative Statics customers, s, and s,, at alternative values of v. Each line represents the willingness to pay Since Section IV indicated that the quali- for z at a given utility index. At the same tative results are not affected, it is conven- value of v the functions are parallel and s, ient to exploit the assumption of common s type consumers outbid s2 types at all values in the consuming population. Demand and of z. In equilibrium the relevant v (the nega- supply shlfts are considered in turn. tive of the utility index) for type s2 must exceed that for type s,. Otherwise the former A. Demand Shifts group would not purchase the service at all. Consequently the observed market relation An increase in the number of consumers must be the envelope of functions such as or in the intensity of their demands for y p=v,z-s, andp=v2z-s2, the heavier curve increases the market demand for services. in Figure 3. The envelope is convex. Evi- Market equilibrium price v rises due to rising dently the main features of the analysis above supply price. Hence unit prices, p(q), of all hold for each linear piece of p(z) in Figure 3. sellers increases. Since R(q) increases every- There are, however, additional implications where, less talented people enter. At the same of sorting between segments. First, the time, existing sellers expand their scales of more talented sellers gravitate to that seg- operations. Though average quality of sellers ment of the market with the largest value falls, all previous entrants earn larger rents of v, precisely the reason why the con- than before, and the largest increases accrue vexity implications of the previous analysis to the most talented persons (see the effect of are strengthened. Second, consumers with v in equation (13) or (18)). Therefore the smaller values of s buy from less talented distribution of reward becomes more skewed sellers. This is quantity-quality substitution than before. at work: buyers with smaller values of s find The important practical implication is that quantity relatively cheaper and economize it is monetarily advantageous to operate in a on quality, while those with large values of s larger overall market; and it is increasingly demand greater quality and economize on advantageous the more talented one is. No quantity. Adding more types of consumers smooths the locus of equilibrium points in Figure 3 without affecting the general prin- the most talented. This raises subtle questions of the ciple~.'~ definition of markets that remain to be solved. An approximate solution in the analysis here is to adjust the density of +(q):if several sellers are thought by differ- "~eder points out that the market is less con- ent customers to have the same value of q, that is nearly centrated if there are differences of opinion on who is the same as more mass in + at that value. 856 THE AMERICARi ECONOMIC REVIEW DECEMBER 1981 wonder that the best economists tend to be the costs of congestion still can imply in- theorists and methodologists rather than nar- creases in both R(q) and R1(q).However, row field specialists, that the best artists sell that is a less likely outcome than when de- their work in the great markets of New York mand is elastic. and Paris, not Cincinnati, or that the best The practical importance of all this is re- writers are connected with the primary liter- lated to technical changes that have in- ary centers such as New York or London. creased the extent of scale economies over The best doctors, lawyers, and professional time in many activities. Motion pictures, ra- athletes should be found more frequently in dio, television, phono reproduction equip- larger cities. For a given place in the distri- ment, and other changes in communications bution of talent, it is more lucrative to be a have decreased the real price of entertain- violinist than an accordianist, a heavyweight ment services, but have also increased the than a flyweight, a rock musician than a folk scope of each performer's audience. The ef- singer, a tennis player than a bowler, or a fect of radio and records on popular singers' writer of elementary texts rather than of incomes and the influence of television on monographs. the incomes of news reporters and profes- sional athletes are good cases in point. And B. Supply Shifts there are fine gradiations within these cate- gories. Television is evidently a more effec- The interesting experiments are changes in tive medium for American football and internal and external diseconomies. Lesser basketball than it is for bowling, and in- diseconomies increase the market supply of comes reflect it. Nonetheless, television has services, reduce the equilibrium value of v, had an enormous impact on the incomes of and make consumers better off. The effects the top bowlers, golfers, and tennis players. on the distributions of talents and rents are because their markets have expanded. The less obvious and complicated by the presence "demise" of the theatre is more a complaint of two opposing forces: the reduction in v about competition from the larger scale lowers unit prices of all sellers, tending media; and incomes of the top performers in to decrease individual output and reward; the theatre, motion pictures, and television whereas the reduction in costs or congestion certainly are closely geared to audience tends to increase them. The balance between size. These changes are not confined to the the two depends on the elasticity of demand entertainment sector. Undoubtedly, secular for services. changes in coinmunications and transpor- If demand for services is sufficiently elas- tation have expanded the potential market tic, then cost reducing effects swamp the for all kinds of professional and information decline in unit prices and rents of sellers services, and allowed many of the top practi- increase. The rent-talent gradient increases tioners to operate at a national or even inter- as well and there is greater concentration in national scale. With elastic demands there is the distribution of rewards among the most a tendency for increasing concentration of talented. A reduction in the internal disecon- income at the top as well as greater rents for omy induces entry at the extensive margin, all sellers as these changes proceed over time. and the average seller becomes less talented. However, a reduction of the external dis- C. Interactions economy, if large enough, can actually re- duce the number of sellers, kicking out the A change in s shifts the supply of services, less talented and increasing the average qual- not demand, even though it is a consumer ity of those remaining. If demand is inelastic, parameter. This has no counterpart in stan- then the number of sellers declines and, since dard theory. Demand is not directly affected those leaving are selected from the lower tail, because c embodies all relevant information the average remaining talent rises. Effects for the consumption decision. Supply is on the return function R(q) are ambiguous shifted because s affects unit prices (see (4)). in this case, though sufficient reductions in An increase in s reduces unit prices at any VOL. 71 NO. 5 ROSEN: ECONOAfICS OF SUPERSTARS 857 value of u and reduces market supply. There- nius to undertakings vaster, and ex- fore the equilibrium service price u increases tending over a wider area, than ever and the rent distribution is altered in favor before. of the more talented sellers. The less talented It is the first cause.. . that enables leave the market. Both the increase in aver- some barristers to command very high fees, for a rich client whose reputation, age quality of sellers and greater concentra- or fortune, or both, are at stake will tion in rewards at the top reflect customers' scarcely count any price too high to substitution of quality for quantity as s rises. secure the services of the best man he Since important components of s are time can get: and it is this again that enables and effort costs, time-series changes are cor- jockeys and painters and musicians of related with consumer earnings. Therefore exceptional ability to get very high market demand increases at the same time prices.. . . But so long as the number of that supply is reduced, resulting in an even persons who can be reached by a hu- greater increase in o and additional skew. It man voice is strictly limited, it is not can even push the extensive margin down very likely that any singer will make an advance on the £10.000 said to have rather than up. The incentives for invest- been earned in a season by Mrs. ments in time saving innovations that tend to Billington at the beginning of the last reduce s as earnings rise, for example, con- century, nearly as great as that which sumption at home, have been well remarked the business leaders of the present gen- upon in the literature.13 eration have made on those of the last. [pp.685-861 VI. Conclusion Even adjusted for 1981 prices, Mrs. In discussing the general influence of eco- Billington must be a pale shadow beside nomic progress on value, Alfred Marshall Pavarotti.I4 Imagine her income had radio wrote: and phonograph records existed in 1801! What changes in the future will be wrought The relative fall in the incomes to be by cable, video cassettes, and home com- earned by moderate ability.. . is ac- puters? centuated by the rise in those that are obtained by many men of extraor- I4The entries for Elizabeth Billington in the eleventh dinary ability. There never was a time edition of the Enqclopediu Rritunnicu and GrmeS Musi- cal Dictionury indicate that she earned somewhere be- at which moderately good oil paintings tween £10.000 and £15,000 in the 1801 season singing sold more cheaply than now, and . . . at Italian Opera in Covent Garden and Drury Lane. She is which first-rate paintings sold so dearly. reported to have had an extraordinary voice and was A business man of average ability and highly paid throughout her professional life. but there is average good fortune gets now a lower a hint that the 1801 sum was unusual even for her. No rate of profits.. . than at any previous information is given on endorsements. time, while the operations, in which a man exceptionally favoured by genius and good luck can take part, are so REFERENCES extensive as to enable him to amass a large fortune with a rapidity hitherto G. Becker, "A Theory of the Allocation of unknown. The causes of this change are two; Time," Econ. J., Sept. 1965, 73, 493-508. firstly, the general growth of wealth, -, "A Theory of Marriage: Part I," J. and secondly, the development of new Polit. Econ., July/Aug. 1973, 81, 8 13-46. facilities for communication by which E. Berglas, "On the Theory of Clubs," Amer. men, who have once attained a com- Econ. Rev. Proc., May 1976, 66, 116- 121. manding position, are enabled to apply , and D. Pines, "Clubs as a Case of their constructive or speculative ge- Competitive Industry with Goods of Vari- able Quality," Tel Aviv University, 1980. J. M. Buchanan, "An Economic Theory of "see Becker (1965) Clubs," Economics, Feb. 1965. 32, 1-14. 858 THE AMERICAN ECONOMIC RE VIEW DECEMBER I981 M. Friedman, "Choice. Chance and the Per- Size Distribution," in Lee Soltow, ed.. Six sonal Distribution of Income," J. Polit. Papers on the Size Distribution of Wealth Econ., Aug. 1953, 61. 277-90. and Income, New York: Columbia Univer- D. Grubb, "Power and Ability in the Distribu- sity Press, 1969. tion of Earnings," Centre for Labour Eco- Albert Rees, The Economics of Work and Pay, nomics. London School of Economics, New York: Harper and Row, 1973. 1980. S. Rosen, "Hedonic Prices and Implicit E. Lazear and S. Rosen, "Rank-Order Tourna- Markets: Product Differentiation in Pure ments as Optimum Labor Contracts," J. Competition," J. Polit. Econ., Jan./Feb. Polzt. Econ., 1981. 1974, 82, 34-55. Harold Lydall, The Structure of Earnings, Ox- , "Substitution and Division of La- ford 1968. bor," Economica, Aug. 1978, 45, 235-50. Alfred Marshall, Principles of Economics, 8th M. Sattinger, Capital and the Distribution ed., New York: MacMillan, 1947. of Labor Earnings, Amsterdam: North- Jacob Mincer, Schooling, Experience and Holland, 1980. Earnings, New York: Columbia University M. Trajtenberg, "Aspects of Consumer De- Press, 1974. mand for Characteristics," Maurice Falk M. Reder, "A Partial Survey of the Income Institute, Jerusalem, 1979. http://www.jstor.org LINKED CITATIONS - Page 1 of 2 - You have printed the following article: The Economics of Superstars Sherwin Rosen The American Economic Review, Vol. 71, No. 5. (Dec., 1981), pp. 845-858. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28198112%2971%3A5%3C845%3ATEOS%3E2.0.CO%3B2-R This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. [Footnotes] 3 A Theory of Marriage: Part I Gary S. Becker The Journal of Political Economy, Vol. 81, No. 4. (Jul. - Aug., 1973), pp. 813-846. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197307%2F08%2981%3A4%3C813%3AATOMPI%3E2.0.CO%3B2-4 13 A Theory of the Allocation of Time Gary S. Becker The Economic Journal, Vol. 75, No. 299. (Sep., 1965), pp. 493-517. Stable URL: http://links.jstor.org/sici?sici=0013-0133%28196509%2975%3A299%3C493%3AATOTAO%3E2.0.CO%3B2-N References A Theory of the Allocation of Time Gary S. Becker The Economic Journal, Vol. 75, No. 299. (Sep., 1965), pp. 493-517. Stable URL: http://links.jstor.org/sici?sici=0013-0133%28196509%2975%3A299%3C493%3AATOTAO%3E2.0.CO%3B2-N NOTE: The reference numbering from the original has been maintained in this citation list. http://www.jstor.org LINKED CITATIONS - Page 2 of 2 - A Theory of Marriage: Part I Gary S. Becker The Journal of Political Economy, Vol. 81, No. 4. (Jul. - Aug., 1973), pp. 813-846. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197307%2F08%2981%3A4%3C813%3AATOMPI%3E2.0.CO%3B2-4 An Economic Theory of Clubs James M. Buchanan Economica, New Series, Vol. 32, No. 125. (Feb., 1965), pp. 1-14. Stable URL: http://links.jstor.org/sici?sici=0013-0427%28196502%292%3A32%3A125%3C1%3AAETOC%3E2.0.CO%3B2-L Choice, Chance, and the Personal Distribution of Income Milton Friedman The Journal of Political Economy, Vol. 61, No. 4. (Aug., 1953), pp. 277-290. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28195308%2961%3A4%3C277%3ACCATPD%3E2.0.CO%3B2-1 Rank-Order Tournaments as Optimum Labor Contracts Edward P. Lazear; Sherwin Rosen The Journal of Political Economy, Vol. 89, No. 5. (Oct., 1981), pp. 841-864. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28198110%2989%3A5%3C841%3ARTAOLC%3E2.0.CO%3B2-E Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition Sherwin Rosen The Journal of Political Economy, Vol. 82, No. 1. (Jan. - Feb., 1974), pp. 34-55. Stable URL: http://links.jstor.org/sici?sici=0022-3808%28197401%2F02%2982%3A1%3C34%3AHPAIMP%3E2.0.CO%3B2-U Substitution and Division of Labour Sherwin Rosen Economica, New Series, Vol. 45, No. 179. (Aug., 1978), pp. 235-250. Stable URL: http://links.jstor.org/sici?sici=0013-0427%28197808%292%3A45%3A179%3C235%3ASADOL%3E2.0.CO%3B2-6 NOTE: The reference numbering from the original has been maintained in this citation list.