Doshisha University Center for the Study of the Creative Economy
Discussion Paper Series No. 2013-04

Nonlinear Effects of Superstar Collaboration:
Why the Beatles Succeeded but Broke Up

Tadashi Yagi

Discussion Paper Series

1Nonlinear Effects of Superstar Collaboration:
Why the Beatles Succeeded but Broke Up

Tadashi Yagi
(Faculty of Economics, Doshisha University)

Abstract
This paper examines the knowledge creation process that takes place in a group of
superstar innovators. How does a superstar innovator influence the generation ofknowledge in a team? In particular, does the effect increase nonlinearly as the numberof talented team members increases? We explore these questions by considering the
success and subsequent breakup of the Beatles, one of the most popular music groups inhistory. Through the mathematical analysis for exploring the fundamental mechanismof the breakup and its implications for superstar collaborations, we show that the
breakup was an inevitable consequence of the superstar collaboration.

Corresponding Author: Tadashi Yagi
Faculty of Economics, Doshisha University, Kamigyo, Kyoto, 602-8580, Japan
Tel.: +81-75-251-3629
Fax: +81-75-251-3629
E-mail: tyagi@mail.doshisha.ac.jp

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I. Introduction
Studies on the knowledge creation process are becoming increasingly important inregard to the creative economy, where new marketable knowledge and concepts are thekey drivers of value creation and innovation. To date, much of the research in this field
has focused on knowledge management and its effects on innovation, but the creative
process is treated largely as a black box. Levin and Stephan (1991) empirically examine
research motivations and show that the investment motive model is more effective thanthe consumption motive model in explaining the pattern of life-cycle research activities.
Weitzman (1998) establishes the micro-foundations of knowledge creation, finding thatthe process entails not so much generating new ideas, as recombining an abundance ofold ideas into a new usable form. Jones (2009) investigates recent trends intechnological progress by considering the increasing educational burden on innovators,
suggesting that increasing difficulty of innovation hinders long-run economic growth.
In the framework of knowledge creation, how are new ideas created within a team?
Wuchty, Jones, and Uzzi (2007) point out the benefits of freely sharing knowledge
through collaboration, and Mas and Moretti (2009) empirically study the strength of thepeer effect within a group, showing that the productivity of an individual worker affects
the productivity of co-workers, even for low-skilled workers.
This paper shares the similar idea with Azoulay, Zivin and Wang (2010) which
estimate the magnitude of spillovers generated by academic “superstars”. This examines
the knowledge creation process that takes place in a group of superstar innovators. How
does a superstar innovator influence the generation of knowledge in a team? In
particular, does the effect increase nonlinearly as the number of talented team members
increases? We explore these questions by considering the success and subsequent

3

breakup of the Beatles, one of the most popular music groups in history.
Of the four musicians in the Beatles—John Lennon, Paul McCartney, George
Harrison and Ringo Starr—Lennon has a reputation for being innovative, sociallyconscious, and knowledgeable of aesthetics; McCartney is known for his innovative andjoyous approach to music; Harrison is regarded as the quiet and philosophical Beatle
with a strong sense of aesthetics; and Starr is known as the skillful, innovative,
good-natured drummer of the band. With four distinct and complementary personalities
brought together in a creative and synergistic alliance, the Beatles quickly found successand made music that was distinguished for being innovative, joyful, appealing, anddiverse.
Other talented collaborators who contributed to the success of the Beatles arefollowing two persons. Producer George Martin played a leading role in creating thedistinct sound of the band by overseeing the group’s musical arrangements, and the
band’s manager Bryan Epstein (who died in 1967) had a keen aptitude for promotion.
By examining the way the Beatles created their music, we aim to examine whether anonlinear effect of superstar collaboration is generated, and if so, how the effect comes
about; moreover, we seek to elucidate the conditions for successful superstar
collaboration.
In considering the nonlinear effect of collaboration in the Beatle, we cannot ignore
the fact that the Beatles broke up only eight years after releasing their debut record.
While there have been many theories regarding the breakup of the band, this paper
focuses on the fundamental mechanism of the breakup and its implications for superstar
collaborations. We show that the breakup was an inevitable consequence of thesuperstar collaboration. We present results of an analysis using a mathematical model to4examine when collaboration becomes no longer beneficial for individual members of
the Beatles, and show that the breakup of a superstar collaboration is a logicalconsequence of the knowledge creation process.
This paper is organized as follows. Section II describes the knowledge creation
process of the Beatles, illustrating the nonlinear effects of superstar collaboration.
Section III presents a mathematical model for capturing the breakup mechanism of
superstar collaborations. Section IV discusses the implications of the findings for theknowledge creation process in general.
II. Knowledge Creation Process of the Beatles
This section describes the knowledge creation process of the Beatles and illustrates thenonlinear effects of collaboration between talented music artists. For this purpose, we
focus on cases of co-creation documented in journals, books, and videos.
A. Definition of Nonlinearity
The quality of music created by the Beatles is described by several factors (denoted byei ) such as aesthetics, social consciousness, innovativeness, joyfulness, philosophicaldegree, harmoniousness, and variety. The linear characteristic equation of theco-creation process is defined asQ(λ e1 , λ e2 , λ e3 ,..., λ en ) = λQ(e1 , e2 , e3 ,..., en ) ,(1)

where Q is the quality of music created. If the quality function is specified by the linear
combination of elements

Q(e1 , e2 , e3 ,..., en ) =a0 + a1e1 + a2 e2 + a3e3 ,..., an en ,(2)

the quality of music will increase linearly as the number of talented artists in the group5

increases.
On the other hand, the nonlinear characteristic equation of co-creation process isdefined asQ(=
λ e1 , λ e2 , λ e3 ,..., λ en ) λ t Q(e1 , e2 , e3 ,..., en ), t > 1 .

(3)One example of a specification that exhibits nonlinear characteristics is multiple
combination of quality elements:n

Q(e1 , e2 , e3 ,..., en ) = ∏ ai ei .

(4)

i =1

The analysis aims to ascertain whether the quality of music increases nonlinearly as thenumber of talented artists in the group increases.

B. Understanding Nonlinearity
Before examining the knowledge creation process, it is important to have an intuitive
understanding of nonlinearity. Connolly and Krueger (2006) and Strobl and Tucker(2000) argue that there is a skewed distribution of revenue from album sales. This is inline with the “superstar effect” proposed by Rosen (1981). In other words, the sales
revenue differs drastically among music groups; a linear correlation between album
sales and album quality implies that the quality of albums differs drastically amongrecording artists. Notably, there is strong evidence showing that nonlinear effects arise
from the collaboration of superstar innovators as seen from the sales revenue generated
by the Beatles as a group and generated by individual Beatles members after the band
split.
According to the website Statistic Brain, as of April 16, 2013, the estimated total
number of albums sold by the Beatles is 2.3 billion. Online encyclopedia Wikipedia

6

estimates the total number of albums sold by McCartney and Lennon as solo artists to
be around 100 million and 14 million, respectively. These figures suggest strong
nonlinear effects arising from superstar collaboration in the band.

Table 1

ALBUM SALES OF THE BEATLES
Data
Total album sales statistics
Total albums sold2,303,500,000

Total albums sold on iTunes

585,000

Total singles sold on iTunes

2.8 million

Billboard chart statistics
Total weeks on chart

1,278 weeks

Total number ones

15

Total weeks at number one

175 weeksAlbum with longest time spent at number one

30 weeks(Please Please Me)
Source: http://www.statisticbrain.com/the-beatles-total-album-sales/.
Note: As of April 16, 2013.Statistics are from the Recording Industry Association ofAmerica, Apple Records, EMI. Date verified is July 23, 2012.

The nonlinear quality improvement in the quality of the Beatles’ music can be
attributed to several factors. Foremost, the appeal of the band lies in the evolution of the7group’s musical compositions. With each new album produced in an unprecedented
lyrical and musical style, the Beatles created music in a way which resembled themaking of art. A good example is the eclectic Abbey Road, a bestselling album whichbrings together elements of rhythm and blues, pop, soul, and rock and roll. By
experimenting with new tunes and musical arrangements, the band improved on thesound and quality of its music. This would not have been possible if the band were notmade up of talented artists successfully collaborating with one another.
By contrast, during the early years of the Beatles when Starr had yet to join the band,the sound of the group was characterized by a relatively lower quality (The Beatles
Anthology, 1995). This was during the period from 1956 to 1962 when Pete Best, who
was widely thought to be less talented than Starr, was still the drummer of the band. The
implication is that artists whose skills are not on par with their group members canadversely affect the quality of the team output.
This is also evidence revealing the nonlinear properties of knowledge creation. Basedon (4), we can write the following:=Q(e1 , e2 , e3 ,.., 0,..,
en )

n

=
ai ei 0 .
∏

(5)

i =1

The equation shows that with the presence of a subpar element, the overall quality ofmusic created is severely affected, even reduced to zero in the extreme case.

C. Creative Process of the Beatles
This section will examine the creative process in which superstar innovators engage in.
There is no doubt that the Beatles were a group of extraordinarily talented musicians.
Collaboration was key to the band’s success. For instance, Lennon and McCartney were8

writing partners and co-wrote most of the band’s bestselling hits. 1 In the recording
studio, the four band members, together with Martin, regularly exchanged ideas andworked on enhancing the quality of their songs. This is evident based on footage in The
Beatles Anthology (1995). The liner notes from the Beatles’ album Sgt. Pepper’s Lonely
Hearts Club Band (1995) also provides revealing insights into the recording sessions of
the band:The Beatles’ musical ideas progressed in a most tangible way with eachalbum they recorded. Geoff Emerick, the recording engineer who with
George Martin formed the imaginative team which translated the Beatles’
requirements onto tape, once totted up the number of hours put into themaking of Sgt. Pepper and came up with 700. Please Please Me, theBeatles’ first album, was recorded in 585 minutes.
“The Beatles insisted that everything on Sgt. Pepper had to be different,”
says Emerick, “so everything was either distorted, limited, heavily
compressed or treated with excessive equalization. We had microphones
right down the bells of the brass instruments and headphones turned into
microphones attached to violins. We plastered vast amounts of echo onto
vocals, and sent them through the circuitry of the revolving Leslie speakerinside a Hammond organ. We used giant primitive oscillators to vary the
speed of instruments and vocals and we had tapes chopped to pieces andstuck together upside down and the wrong way round.” (Sgt. Pepper’s
Lonely Hearts Club Band 1995, page 1. )1 In interview footage from The Beatles Anthology (1995), McCartney said he and Lennon composed
songs together when on concert tours.

9

The creative process which went into the making of the Beatles’ songs was notwithout challenges. According to Emerick, Geoff; Massey, Howard (2006) andLewisohn (1988), more than 70 takes were required in order to record one song.

For

example, in recording “Ob-la-di Ob-la-da” from The Beatles (a.k.a. The White Album),
Lennon played the song’s piano countless times because McCartney was unsatisfied
with it.
No doubt, the caliber and talent of each band member has a direct bearing on thequality of music produced by the group. Bands made up of individually talented artists
are more likely to produce better music as a result of the regular exchange of ideas
within the group. Former Apple CEO Steve Jobs was reportedly impressed with theperfectionism of the Beatles after listening to a bootleg of “Strawberry Fields Forever,”which recorded the series of many number of takes of “Strawberry Fields Forever”(Isaacson, 2011). Indeed, talented artists are also more likely to spot issues and engage
in problem solving, which would improve the quality of music created.
Another factor key to successful creative collaboration in a team is the need for equal
member status. Take for example a band with a chief songwriter who has a dominantposition within the group. In such a band, the less dominant members are unlikely tocomment on aspects relating to musical composition. Consequently, the caliber of the
songwriter becomes the determining factor of the quality of music produced by the band.
The implication is that the positive collaborative effect in the band becomes greatly
weakened, and that the quality of the music might not be markedly improved during therecording sessions.
Hence the quality of the knowledge creation process of a group is determined by the
degree of idea exchange and competition within the group. Music groups made up of

10

members of unequal status easily foster rivalry. On the other hand, bands with members
who have relatively equal status form highly creative and productive teams. Thesebands thrive on a free exchange of ideas, resulting in the nonlinear effects of superstar
collaboration.
This point is described by the following formulation. Suppose that the quality ofmusic be represented byQ(e1 , e2 , e3 ) = A( p (e1 ; h)e1 )α e2β e3γ ,(6)where e1 is innovativeness, e2 is joyfulness, e3 is philosophical degree, α theweight of innovativeness in producing quality music, β is the weight of joyousness,and γ is the weight of philosophical degree. Let p (e1 ; h) be the probability ofsuccess in innovation which depends on the level of competitiveness between members
within the group, h . The nonlinear effects of superstar collaboration emerge when
there is mutual cooperation in a group, increasing the probability of successful
innovation. Lennon stated in an interview that he strived to create fundamentally
innovative music during the Sgt. Pepper’s Lonely Heart Club Band recording session in
response to the creative efforts of McCartney and Harrison (Dig, 2010). Especially, theuse of discord was one of his innovative challenges. The success of the innovative
masterpiece “I am the Walrus” on the album Magical Mystery Tour is the fruit ofcontinuously striving to include a discord in the music.

D. The Breakup of the Beatles
The Beatles had a critical year in 1967.Lennon met Yoko Ono, an avant-garde artist who
was widely believed to be the catalyst leading to the split of the band. While theappearance of Ono was not the main reason that caused the breakup of the Beatles, the11reasons why Lennon needed Ono and Harrison needed Indian philosophy were quite
important. 2 That year also marked the time when Lennon began to take an interest in theavant-garde movement. 3 He would soon incorporate elements of avant-garde into themusic of the Beatles for the band’s next album. “Revolution 9” from The White Album
was an experimental sound collage and shows Lennon’s style diverging from that of theother Beatles. Meanwhile, behind the scenes in the recording studios, trouble wasbrewing for the Beatles. In another round of footage from The Beatles Anthology (1995),
Martin noted that there was much unhappiness among the band members during themaking of the album Let It Be, and the worsening relationships among the band
members is evident in the film Let It Be.
As the four musicians began to explore different musical directions, the positivesynergistic effects of collaboration in the Beatles began to fade. 4 At the same time, there
was a change in the power dynamics of the band. McCartney began taking on an
increasingly dominant role in the Beatles, a factor which also catalyzed the split of theband, along with Harrison’s complaint of unequal treatment. The following section
explores the breakup of the Beatles by using a mathematical model.

III. The Breakup of the Beatles as a Knowledge Creation Model
Earlier in the paper, there is an analysis of the factors resulting in the nonlinear effectsof superstar collaboration. In this section, the breakup of the Beatles will be analyzed

2

Lennon’s “Imagine” and Harrison’s “My Sweet Road” are proof that Ono and Indian philosophy were
important to them.
3 Lennon said in an interview published in The Dig Special Edition that he did not agree with all of thetenets of the avant-garde movement. What he had wanted to do was create something similar toavant-garde art.
4
The last album recorded by the Beatles, Abbey Road, was considered a masterpiece. It was also
revealed in footage from The Beatles Anthology that the band members worked with each other in an
atmosphere of positive collaboration during the making of the final album.

12

using a knowledge creation model.
Let kie (t ) denote the knowledge stock of individual i at time t. For this case study,
e may represent joyousness, estheticism, social consciousness, philosophical degree andinnovativeness. The talent of an artist is a function of his or her knowledge
accumulation. As human capital matures, knowledge creation within the groupdecreases for two reasons: decreasing differences in knowledge levels among members
and reduced flexibility in accepting differences of knowledge creation. The knowledge
accumulation function can be written asdk= f (ki )f (α ∑ d (ki − k j )) + g (ki )f ((1 − α ) ∑ d (ki − k j )) .
dt
j∈M
j∉M

(7)

Here, f represents the degree of flexibility in accepting differences of knowledge
creation and is a decreasing function of k, and φ is an increasing function. Let g be theknowledge creation which arises from collaborating with a non-member, assuming that
it is an increasing function of k. Let ki (t ) denote the vector of knowledge of member i
at time t and let α (t ) denote the portion of time spent with the member of the group.
The members control the portion of time they spend working together so as tomaximize their creative output in terms of quality, subject to the dynamic equation ofknowledge capital k. The objective function is represented by (8) and the dynamic
equation of knowledge is shown by (9). Assume the time discount rate is zero, withequal weight given to each period of time so as to simplify the model:T

Maxα ∫ Q(ki (t ) : k j (t ))dt ,

(8)

t

dk
s.t.
f (ki )f (α ∑ d (ki − k j )) + g (ki )f ((1 − α ) ∑ d (ki − k j ))
=
dt
j∈M
j∉M
,

(9)

13



=
H Q(ki (t )) + λ  f (ki )f (α ∑ d (ki − k j )) + g (ki )f ((1 − α ) ∑ d (ki − k j )) 
j∈M
j∉M



, (10)The optimality conditions are given by (11) and (12), as follows:∂H= λ[ f (ki )f '(α ∑ d (ki − k j ))�( ∑ d (ki − k j ))
∂α
j∈M
j∈M

,

0
− g (ki )f '((1 − α ) ∑ d (ki − k j ))�( ∑ d (ki − k j ))] =
j∉M

(11)

j∉M

∂H
∂Q
dλ
=
−
=
−
− λ[ f '(ki )f (α ∑ d (ki − k j )) + f (ki )f (α ∑ d '(ki − k j ))
∂ki
∂ki
dt
j∈M
j∈M
+ g '(ki )f ((1 − α ) ∑ d (ki − k j )) + g (ki )f ((1 − α ) ∑ d '(ki − k j ))]j∉M. (12)j∉M

The transversality condition islim l (t )k (t ) = 0 .

(13)

x →T

As long as λ is positive, the optimality condition (11) implies that the optimal timeshare α is determined at the point where the following equation is satisfied:f (ki )f '(α ∑ d (ki − k j ))�( ∑ d (ki − k j ))
j∈M

j∈M

= g (ki )f '((1 − α ) ∑ d (ki − k j ))�( ∑ d (ki − k j ))
j∉M

(14)

j∉M

Since f ' is a decreasing function, the left-hand side of the equation is a decreasing
function of α while the right-hand side is an increasing function of α . The left-handside can be considered the marginal productivity of within group knowledge creation(MPI) and the right-hand side measures the marginal productivity of outside groupknowledge creation (MPO). Note the relationship between ki and the optimal level of

α . To illustrate this, we take the first derivative of the left- and right-hand sides withrespect to ki . The first derivative of the left-hand side is given by

14∂f (ki )f '(α ∑ d (ki − k j ))�( ∑ d (ki − k j ))j∈Mj∈M
= f '(ki )f '(α ∑ d (ki − k j ))�( ∑ d (ki − k j ))
∂ki
j∈M
j∈M
+ f (ki )f ''(α ∑ d (ki − k j ))�α ∑ d '(ki − k j )�( ∑ d (ki − k j ))
j∈M

j∈M

(15)

j∈M

+ f (ki )f '(α ∑ d (ki − k j ))�( ∑ d '(ki − k j )).
j∈M

j∈M

The derivative of the right-hand side with respect to ki is∂g ( ki )φ '((1 − α ) ∑ d ( ki − k j ))�( ∑ d ( ki − k j ))
j∉M

j∉M

∂ki

=

g '( ki )φ '((1 − α ) ∑ d ( ki − k j ))�( ∑ d ( ki − k j ))
j∉M

(16)

j∉M

+ g ( ki )φ ''((1 − α ) ∑ d ( ki − k j ))�(1 − α ) ∑ d '( ki − k j )�( ∑ d ( ki − k j ))
j∉M

j∉M

j∉M

+ g ( ki )φ '((1 − α ) ∑ d ( ki − k j ))�( ∑ d '( ki − k j )).
j∉M

j∉M

Because f is a decreasing function of ki and g is an increasing function of ki ,, theMPO and MPI curves shift as ki increases, as shown in Figure 1. Therefore, one canconclude that the amount of optimal time spent collaborating within the group decreases.
This suggests that the Beatles broke up as the knowledge stock of its members rose.

MPO(t1)MPI

MPO(t0)MPO

MPI(t0)MPI(t1)

α(t1)

α(t0)

15

Figure 1: Value of co-creation within and outside group

Combined with (6), the optimality conditions imply that the quality of music
produced by the group is as follows:

Q = Q( A( p (e1 (k1 (α1 ),.., k4 (α 4 )); h)e1 (k1 (α1 ),.., k4 (α 4 )))α e2β (k1 (α1 ),.., k4 (α 4 ))e3γ (k1 (α1 ),.., k4 (α 4 ))(17)
Based on this equation, one can finally understand why the final album recorded by the
Beatles, Abbey Road, is critically acclaimed as a masterpiece, even though it wasrecorded when the group was on the verge of breaking up: the period during which thealbum was recorded coincided with the creative peak of each member of the band. As
opposed to the recording of Let It Be, which was made at a time marked by tense
relations between the band members, the production of Abbey Road saw the Beatles
successfully work as a team.
IV. Conclusion
Taking up the case of the Beatles, this paper examined the knowledge creation processof superstar innovators and the nonlinear effects which arose from peer collaboration.
The findings of this paper may be applied to the practice of human resource
management in firms.
An important implication of this work is that the incentives for collaborating within ateam tend to decrease as the knowledge of each member is shared over time.
Company executives may decide to employ the same personnel so as to maintaincontinuity of the company brand and image or increase the staff turnover as a way of
inculcating a more competitive spirit in employees.
The paper also revealed the conditions conducive to successful innovation through16

collaboration. A harmonious environment in which team members can trade ideas freely
and engage in fair play fosters creativity, leading to innovation. This finding will be
useful to businesses for planning the organizational structure of the company’s researchand development division. In particular, the study will be useful for businesses in
deciding on a flat or hierarchical structure for the research team. It is implied that overly
strong initiative on the part of an individual team member in the research anddevelopment process possibly deteriorates the collaboration effect among researchers.
Lastly, the analysis illustrated the importance of diversity during the process ofknowledge creation. The nonlinear effects of superstar collaboration emerge in a teammade up of talented individuals who each possesses a unique specialist knowledge orexpertise. By putting to use the diverse sets of skills and knowledge, talented innovators
can achieve successful innovation in a team, and the value of the innovation and new
knowledge increases nonlinearly as the talent of members of the team increases.

17

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