Exploring a better design of copyright law
Shinya Kinukawa∗
June, 2015

Abstract
This study proposes a simulation model to explore how the copyrightlaw should be designed; that is, a desirable combination of copyrightlength and breadth. The model also incorporates the cumulative featuresof content creation: the cost of creating new work decreases when larger
parts of past works becomes freely available. The simulation results indicate that copyright protection should first be broad because social welfare
is larger under broad protection than under narrow protection regardlessof length. Then, short length is more desirable because it encourages
creators to publish more works by largely reducing creation cost. Thus,
for both social welfare and cultural prosperity, short and broad copyrightwould be the most desirable, and long and narrow copyright would be theleast desirable.
Keywords: Copyright length, Copyright breadth, Simulation∗ Associate

Professor, Faculty of Global Media Studies, Komazawa University, E-mail:kinukawa@komazawa-u.ac.jp

1

Introduction

People are discontent with the current copyright law, and disagreement between
right holders and users has been growing. A recent example of such disagreement
is seen in the Trans-Paciﬁc Partnership (TPP) negotiations. Negotiators under
the inﬂuence of large right-holding companies are proposing to extend copyrightterms by another 20 years beyond the minimum copyright term (50 years) under
the current TRIPS agreement, while many non-government groups including
Creative Commons are opposing it.1
Several economists have tried to mitigate this situation by analyzing an optimal level of copyright protection, focusing only on either its breadth (e.g.,
Novos and Waldman, 1984; Landes and Posner, 1989; Miceli and Adelstein,2006), or the length of the copyright term (e.g., Yuan, 2005; Adilov and Waldman, 2012). Unlike for patent design, length and breadth together have not yet
been explored for copyright law.2
Furthermore, most literature on optimal copyright protection focuses on theeﬀect of a policy change on a creator’s proﬁt, which is the incentive to create a
new work. The exception is Landes and Posner (1989), whose model states thata change in copyright protection level aﬀects not only a creator’s proﬁt froma new work but also the cost of its creation. While extending copyright termincreases a creator’s monopolistic proﬁt, it can also increase the cost of creation
because past works are important inputs for the creation of a new work.3
This study integrates these issues and provides a theoretical framework to
examine optimal combination of copyright length and breadth, incorporating
both a creator’s proﬁt from new works and the creation cost. Length is theterm of copyright protection, and breadth is the amount of a protected workthat can be freely used or copied by non-right holders (e.g., fair-use). The model
this study constructs, however, becomes too complicated to be solved analytically. In the model, past works aﬀect future creation cost, but the dynamics ofcreators’ decision-making is not recursive. Since the cost of creation depends on
1 Creative

Copyright

Commons

Term

Blog

Extension

News,

Makes

No

“An

Open

Sense,”

LetterRyan

toMerkley,TPP
July

Negotiators:
9th,(http://creativecommons.org/weblog/entry/43256, last accessed on April 17th, 2015).
2 For the literature on optimal patent design, see a survey by Rockett (2010)
3 For real world examples of this, see Lessig (2004).

1

2014

whether past works are protected, time itself enters the decision-making problem under ﬁnite copyright terms. Thus, instead of analytically solving for the
optimal length and breadth, the eﬀects of changes in copyright design (lengthand breadth of protection) are examined through numerical simulation.
The simulation is run under the following four sets of copyright design: (i)long length and broad breadth, (ii) long length and narrow breath, (iii) shortlength and broad breadth, and (iv) short length and narrow breadth. The simulation results reveal that on average, the choice of copyright length marginallyaﬀects social welfare. On the other hand, the choice of copyright breadth signiﬁcantly aﬀects social welfare, which is larger under broad protection. Length,
however, aﬀects entry and exit of creators, the number of publications, and accumulation of valuable works, which are greater under short copyright. Therefore,
short and broad copyright can be the most desirable, and long and narrow copyright can be the least desirable, for both social welfare and cultural prosperity.
The rest of the paper proceeds as follows. Section 2 reviews the literature on
the optimal copyright design. Section 3 explains the model. Section 4 presents
the results of the numerical analysis. Section 5 concludes the study.

2

Literature review on the optimal copyrightdesign2.1

Breadth

In the studies focusing on optimal copyright breadth, breadth is usually interpreted as the extent of protection against copying by non-right holders. Miceli
and Adelstein (2006) provide a simple breadth setting and examine the optimalbreadth. In their model, the quality of a work’s copies ranges from zero to one.
Breadth is deﬁned as the maximum quality of the copies that does not infringe
the creator’s copyright. The highest quality copy is provided only by the work’s
creator, and the creator’s copying cost is smaller than any other copiers’ one.
A consumer’s utility depends on a copy’s quality. All consumers have the same
marginal beneﬁt of quality, but they diﬀer in their maximum desired quality.
Introducing legal copies such as fair-use does not aﬀect the creator’s proﬁt if all

2

the consumers buying the highest quality copy are dissatisﬁed with the qualityof legal copies. Based on these model setting, they analyze the relationshipbetween the optimal breadth (fair use) and copying cost.
In Novos and Waldman (1984), the breadth variable aﬀects both a work’s
quality and consumers’ copying costs. Consumers decide whether to buy awork’s copy from the monopolistic provider or to borrow it from a secondary
source for making a copy. The breadth variable aﬀects a consumer’s copying
cost. When the extent of copyright protection increases, more consumers buy
copies from the monopolistic provider, which increase the provider’s proﬁt andthe incentive to invest in a work’s quality. Assuming that consumers’ copying
costs diﬀer across individuals, Novos and Waldman (1984) demonstrate that
depending on the distribution of the copying costs, an increase in copyrightprotection can decrease the social welfare loss due to a work’s low quality. They
further demonstrate that total copying costs for all consumers who borrow andcopy does not necessarily increase when the copyright protection level increases
because consumers with high copying costs shift from borrowing to buying.
Landes and Posner (1989) oﬀer the most comprehensive analysis of breadth.4
In their model, a work’s copies are provided not only by the work’s creator
but also by the copier. Breadth aﬀects the quantity of the copies provided by
the copier. Moreover, breadth aﬀects the cost of creation, which is diﬀerent
across creators, because a creator often borrows expressions of other creators’
works in creating a new work. Thus, the supply of works, which is equal to thenumber of creators providing new titles, rises until the creation cost of a marginal
creator equals the revenue from a work’s copies. By solving the optimal levelof copyright protection to maximize social welfare, Landes and Posner (1989)
drive several implications, one of which states that the protection level should
be higher for a work producing a larger total surplus.

4 Landes and Posner (1989) explain that their copyright protection level variable can include

the length of copyright term. However, their theoretical model is static. In their model, lengthis not distinguished from breadth, and the dynamic process of creators’ behavior is ignored.

3

2.2

LengthYuan (2005) takes a simple approach to analyze the optimal copyright length:
summing each period’s total surplus from the ﬁrst to the last periods. The
copyright length, which is set between the ﬁrst and last periods, determines
the term in which a creator is able to set the monopoly price for a work’s
copies. Several creators exist, and they compete in the markets for copies. The
demand for a work’s copy depends not only on its price but also on the pricesof other works’ copies. This makes the demand function complicated enoughthat an explicit analytical solution cannot be found, and thus, Yuan (2005)
employs a numerical method to solve the optimal length problem. He ﬁnds thatthe optimal length is an increasing function of copying cost and a decreasing
function of demand level. Therefore, short copyright length is socially optimal
when copying cost decreases and the demand for copies increases. These results
are obvious from the model’s assumption that the copyright holders fully exploit
their works. Under the assumption, small copying costs and a large demand levelmean that the copyright holder earns suﬃcient proﬁt to incentivize investment
in a work even when the copyright term is short.
Adilov and Waldman (2013), on the other hand, conclude that inﬁnite copyright length can be socially optimal. Their results are derived from a theoreticalmodel where the copyright holder dynamically optimizes the sequence of eachperiod’s ex-post investment during the copyright term. Longer copyright term
can keep incentivizing ex-post investment after a work is published, and consequently consumers gain a larger surplus from the value-enhanced work due to
ex-post investment. Adilov and Waldman (2013) also compare the ﬁxed-length
copyright protection with the indeﬁnitely renewable one and ﬁnd that the latteris better when the returns on ex-post investments are suﬃciently large.5

5 Yuan (2006) also examines indefinitely renewable copyright protection and concludes thatthe fixed length is better. The model is an extension of Yuan (2005), adding renewal fee andignoring ex-post investment.

4

3

Model

3.1

Basic assumptions

In each period denoted by t, there are Nt creators of works such as books,
movies, or songs. As in the model of Landes and Posner (1989), a creator is notdistinguished from a publisher, ﬁlm studio, or music recording company. Boththe marginal and ﬁxed costs of making and distributing copies (or reproductions)
are set to zero, and creators incur only the cost of creating new works (or thecost of expression).
Each creator i decides the number of new works s/he produces (Kit ) at the
beginning of each period. The period of a new work’s publication is denotedby p, and the term of copyright protection (copyright length) is denoted by Tc .
During the copyright term (t − p ≤ Tc ), a work’s creator is the only provider ofcopies of the work and its derivative works. As in Landes and Posner (1989),
all diﬀerent ways to exploit a work are identically treated and called “providing
copies.” Each work is suﬃciently diﬀerentiated, and the markets for diﬀerent
works’ copies are independent.
Let vitkp be period t’s market value of creator i’s k-th work published inperiod p (≤ t), deﬁned as the sum of all consumers’ willingness to pay for
the work’s copies during period t. Since the marginal cost of providing copies
is zero, vitkp equals the consumer surplus that is provided at the competitive
price. Before a work is published, its market value vitkp is unknown and deﬁned
as the following stochastic variable.
 (1 − δ)t−p v
vitkp =
 0with probability θwith probability 1 − θ,

where 0 < θ < 1, 0 < δ < 1, and v > 0. A work vitkp > 0 is a longtime seller,
whose market value decreases at the rate of δ. All other works’ values are set
to zero.
While the above assumption is for simplicity, a little reality can be added by
setting the success probability θ and the rate of a work’s obsolescence δ very low.
With this parameter setting, the assumption may ﬁt the well-known fact that(a) the great majority of copyrighted works never have much market value and(b) a small percentage of titles constitutes a large share of sales of copyrighted
5materials (Liebowitz and Margolis, 2005). Moreover, Liebowitz and Margolis(2005) show that the longevity of best-selling books is quite long (more than 85
years) based on sample titles reviewed in Book Review Digest in the 1920s.
In addition to copyright length Tc , copyright breadth z is another parameterthat controls copyright protection. The breadth parameter 0 < z < 1 determines the extent of a work’s market value that its creator can appropriate. Fora copyright protected work with the value vitkp > 0, consumers pay for only
zvitkp of the copies because (1 − z)vitkp of the copies is freely available to them.
Then, during the copyright term, a creator’s gross proﬁt from a work in a period is αzvitkp , and consumer surplus is βzvitkp + (1 − z)vitkp , where α + β < 1.
(1 − α − β)zvitkp equals dead weight loss. After the copyright term expires, all
the value vitkp goes to consumers.
An example that determines the extent of z is the separation between idea
and expression. Copyright law protects only the latter. Distinguishing “expression” from “idea,” however, can be diﬃcult. For example, a unique story
plot can be a protected “expression” rather than an “idea,” which enlarges thebreadth parameter z. On the other hand, expanding limitations on copyright,
such as reproduction for private use or fair-use doctrine, would make z small.
All existing works of the value vitkp > 0 contribute to new works’ creation
by reducing the cost of expression. Let Ct be the works’ stock at the end of
period t (after publications) that is used for new works’ creation in the next
period t + 1. Then, Ct is deﬁned as
Ct = (1 − δ)Ct−1 +

Nt ∑
Kit
∑

vitkt ,

i=1 k=1

where the depreciation rate is assumed to be equal to the obsolescence rate of
longtime seller works.

3.2

Dynamics of the model

The expectation of period t’s total proﬁt of a creator i at the beginning (before
publication) of the period is given as
E[πit ] = αzE

[K
it
∑

]
vitkt − e(Kit ) + αz

Kip
t−1 ∑
∑
p=1 k=1k=1

6

1[t − p ≤ Tc ]vitkp ,where e(Kit ) is creator i’s cost of creating Kit new titles in period t, and 1[t−p ≤Tc ] is the indicator function taking 1 if t − p ≤ Tc and 0 otherwise. The costof creating Kit new titles, which the creator i has to bear before publication, is
speciﬁed as(e(Kit ) =

1+

z

∑

∑t−1 ∑Kjp
j̸=i

p=1

k=1 1[t − p ≤ Tc ]vj(t−1)kp

)

Ct−1

2
Kit
.

Thus, if works’ stock becomes large, including other creators’ less protected
works, then the cost of creating new works becomes small, depending on thebreadth parameter z. The cost e(Kit ) decreases when z becomes smaller because
greater use of other creators’ protected works becomes freely available.
It is assumed that a creator who earns negative proﬁt in a period exits frompublishing new works in the next period. Then, each period’s proﬁt can be
more important than the total proﬁts during the copyright term. Moreover,a creator’s decision making is not recursive because time itself enters into it.
The cost of creating new works depends on whether other creators’ existing
works are protected, which is diﬀerent by time due to ﬁnite copyright length.
Thus, instead of dynamically optimizing the sequence of publishing new works,a creator is assumed to decide the number of new works published in a period
by maximizing the proﬁt during that period expected before publication.
Let K̃it be the number of period t’s new works by creator i deﬁned as a real
number. In each period t, creator i solves
max E[πit ],K̃it∗and the solution K̃itis given as

αzθv
∗
K̃it
=
2

(
1+

z

∑

∑t−1 ∑Kjp
j̸=i

k=1 1[t − p ≤ Tc ]vj(t−1)kp

p=1

Ct−1

)−1
.

Then, the number of creator i’s new works in period t is deﬁned as the smallest∗∗
integer larger than or equal to K̃it
, i.e. ⌈K̃it⌉.

After publication, the proﬁt πit is determined, and as noted above, creator iexits from providing new works in the period t + 1 if πit < 0. For simplicity, it isassumed that copyright protected works of exiting creators are not appropriatedby other surviving creators and their copies are provided by an entity that does
not publish new works.
7In addition to the exit of creators, the model also incorporates new entries.
The number of new creators who enter in a period t is assumed to be a stochastic
variable whose mean depends on the stock of woks Ct−1 . The distribution ofµ
new entries is speciﬁed as a Poisson distribution whose mean is given as λCt−1,

where λ > 0 and 0 < µ < 1.
Finally, the surplus in a period s, ws , is deﬁned as the sum of creators’ total
proﬁts from all protected works and consumers’ surpluses from all existing works
minus the costs of creating new works in that period. Let Ñt be the number of
all creators in t including the creators that exited before t (Nt is the number ofcreators that publish new works in t). Then,

ws =

Ñt ∑
Kit
s ∑
∑

1[t − p ≤ Tc ]αzvitkp

t=1 i=1 k=1

+

Ñt ∑
Kit
s ∑
∑
[
]
1[t − p ≤ Tc ]{βzvitkp + (1 − z)vitkp } + 1[t − p > Tc ]vitkp
t=1 i=1 k=1

−

Ns
∑

e(Kis ),

i=1

where the ﬁrst, second, and third lines correspond to creators’ proﬁts, consumers’ surplus, and the cost of creating new works in period s, respectively.
The total surplus from s = 1 to T , WT , is given as
WT =

T
∑

ws
,
(1
+
ρ)s−1
s=1

where 0 < ρ < 1 is the discount rate.

3.3

Settings for the numerical analysis

In the aforementioned model, the process determining WT is dynamic andstochastic. As explained above, although a creator solves a static problem ineach period, the decision-making problem depends on a period because of ﬁnite
copyright term. Moreover, a creator’s decision is aﬀected by the results of othercreators’ decisions, which are also determined stochastically. The model is hardto deal with in an analytical way, and thus, the eﬀects of the two copyright
law parameters Tc and z on the total surplus WT are examined by numerical8

simulation.6
Four simulation runs are set for diﬀerent parameter sets of the copyright law
design, and each simulation runs for 200 periods. The parameter sets are (i)long length (Tc = 100) and broad breadth (z = 0.8), (ii) long length (Tc = 100)and narrow breadth (z = 0.4), (iii) short length (Tc = 50) and broad breadth(z = 0.8), and (iv) short length (Tc = 50) and narrow breadth (z = 0.4).
The conditions other than those of copyright law design are the same forall series of simulation runs. In the initial period (t = 1), there are 10 creators(N1 = 10), and each of them possesses one longtime seller work with the residual
copyright terms Tc − 1. The stock of works for creating new works in the
beginning of the initial period is given as C0 = N1 v. The values of all model
parameters other than Tc and z are listed in Table 1.7
Table 1: Simulation parameters

4

ParametersDeﬁnition

Valuesα

Ratio of creator’s proﬁt to a work’s value

0.5

β

Ratio of consumer surplus to a work’s value

0.25

δ

Obsolescence rate of works

0.01

θ

Success rate that a work is longtime seller

0.1µ

A parameter for new entry

0.1

λA parameter for new entry

0.01

ρ

Discount rate for social welfare

0.05

N1

Initial number of creators

10v

Value of longtime seller

500

Simulation results

Each of the four simulation runs was repeated 200 times. Figure 1 displays
the results of the total surplus WT (T = 1, ..., 200) calculated as the average
6 A similar approach is taken by Muller and Pénin (2006), who use a simulation model todescribe the dynamics of innovation networks and the role of open knowledge disclosure.
7 The parameters of creators and consumers surpluses, α and β, are determined by theassumption that the demand curve is linear. Let the demand function for a work’s copies be
pc = a − bx, where pc is the price and x is the quantity. Then, the creator’s (monopolist’s)
2

2

2

profit is a4b , consumers’ surplus is a8b , and the dead weight loss is a8b .

9outcome of the 200 simulations. In the later ﬁgures, the labels “lb,” “ln,” “sb,”and “sn” denote “long and broad,” “long and narrow,” “short and broad,” and“short and narrow” copyright designs, respectively.
The outcomes indicate a clear diﬀerence between broad and narrow breadth:the total surplus is larger for broad breadth than for narrow breadth. On theother hand, the diﬀerence in length has little eﬀect on the total surplus. The
results imply that copyright protection needs to be broad to increase social
welfare, but the choice of length does not aﬀect social welfare.
1000000
900000800000700000600000500000400000300000
lb

ln

sb

sn

200000100000

1
9
17
25
33
41
49
57
65
73
81
89
97
105
113
121
129
137
145
153
161
169
177
185
193

0

Figure 1: Comparison of total surpluses
The above results are caused by discounting the surpluses in later periods ofthe simulation. In the evaluation of creators’ revenues and social loss in multiple
time periods, those in future periods need to be discounted to be comparablewith present ones, as is stated in the seventeen economists’ amicus curiae briefin support of the petitioners in Eldred v. Ashcroft.8 The brief argues that
there is no economic rationality to extend copyright length because a future
proﬁt increase by copyright extension is marginal because of discounting. The
simulations’ results reﬂect such an argument. Only breadth is diﬀerent fromthe ﬁrst to 50th periods. After the 50th period, diﬀerences in the surpluses of
the two length settings are discounted and becomes insigniﬁcant.
Comparing only the total surpluses may not be suﬃcient to evaluate diﬀerent
8 Brief of George A. Akerlof et al. as Amici Curiae in Support of Petitioners at 12, El-dred

v. Ashcroft, 537 U.S. 186 (2003) (No. 01-618).

10

copyright law designs, and thus this study compares the results of three othervariables: the number of creators who publish new works, the number of publications, and the stock of works among the four diﬀerent copyright law designs.
These variables can be proxies for a variety of works and cultural prosperity,
both of which would be appreciated by consumers. As in Figure 1, all values
are the average outcomes of the 200 simulation runs in the following ﬁgures.
First, Figure 2 depicts the dynamics of creators’ exit and entry. Thoughremaining small, diﬀerences between long and short copyright are observed.
13
lb

ln

sb

sn

12.5

12

11.5

11

10.5

1
8
15
22
29
36
43
50
57
64
71
78
85
92
99
106
113
120
127
134
141
148
155
162
169
176
183
190
197

10

Figure 2: Comparison of the number of creators
The number of creators who publish new works tends to be higher for shortlength than for long length in case of broad breadth copyright designs. On theother hand, the number of such creators tends to be higher for long length thanfor short length in case of narrow breadth. When copyright is short and narrow,the revenues from existing works are the smallest, likely creating the largest
number of exits compared to other copyright designs. On the other hand, shortand broad copyright prevents exits more than long and broad copyright does,
although the revenues from existing works are smaller under short length thanunder long length. The result can be explained by an increase in publications
due to a decrease in publication cost after copyright term expiration, which is
depicted in Figure 3.

11

110
lb

ln

sb

sn

100
90
80
70
60
50
40

1
8
15
22
29
36
43
50
57
64
71
78
85
92
99
106
113
120
127
134
141
148
155
162
169
176
183
190
197

30

Figure 3: Comparison of publications
For short and broad copyright, a large and steep shift occurs in the level of
publication numbers after the 50-th period, when the copyrights of the initialperiod’s longtime sellers expire. When copyright is short, past works that keep
value relatively high become freely available after copyright expiration, which
can largely reduce the cost of creation. Moreover, when copyright is broad, large
revenues are expected from publications. Therefore, publishing a large numberof new titles can be optimal for creators under short and broad copyright. Large
cost reductions after copyright expirations are also observed for other copyrightdesigns except for long and narrow copyright. When copyright breadth is narrow
and thus the expected proﬁts from publications are small, long length can bepreferred by creators because it reduces the risk of exit as shown in Figure
2. However, creating new works can be further discouraged by long copyrightbecause it maintains high creation cost.
The dynamics of works’ stocks traces the number of publications (Figure4). The highest cultural prosperity is achieved by short and broad copyright,
followed by long and broad, and then short and narrow copyright. The lowest
cultural prosperity occurs under long and narrow copyright. These results revealthat the length of copyright terms matters when the quality of the market
outcome, i.e., the number of titles available to consumers, is considered. From
both economic and cultural perspectives, short and broad copyright is most12

desirable, whereas long and narrow copyright is least desirable.
400000
lb

ln

sb

sn

350000300000250000200000150000100000
50000

1
8
15
22
29
36
43
50
57
64
71
78
85
92
99
106
113
120
127
134
141
148
155
162
169
176
183
190
197

0

Figure 4: Comparison of the stock of works

5

Conclusion

Copyright protection is important for the provision of many commercial works.
Few people question this role of copyright law. However, there has been a disagreement regarding the designing of copyright law. The length of copyrightterm and the breadth of copyright protection are the key issues. The literature
has focused only on either length or breadth. This study integrates copyrightlength and breadth into a theoretical framework. The model also incorporates
the cumulative feature of content creation: the cost of creating new work decreases when a larger part of past works becomes freely available. This resembles
cumulative innovation models of the optimal patent design literature.
The simulation results reveal that copyright protection should ﬁrst be broad.
Regardless of length, social welfare is greater under broad protection than undernarrow protection. Then, regardless of the breadth, short term is more desirable
than long term. The result diﬀer from that of the optimal patent design literature for cumulative innovation, which states that the same initial investment
incentive is achieved under broad short patent or narrow long patent (Rockett,2010).
Unlike technological innovation, past and future creative works do not nec13

essarily compete with each other, even when the latter borrows both the idea
and the expression of the former. This makes imitation socially desirable because it eﬃciently boosts the variety of titles available to consumers. A realworld example of such an argument is the creation of Walt Disney’s cartoon
Steamboat Willie in 1928, which was created as a direct cartoon parody of asilent ﬁlm Steamboat Bill,Jr. and brought Mickey Mouse to life (Lessig, 2004).
Therefore, copyright protection imposes both a beneﬁt and a cost to creators.
Short and broad copyright can be the most desirable because it can largely increase exploitation from contemporary works and largely reduce creation costby freeing relatively valuable past works. On the other hand, long and narrow
copyright can be the least desirable because it does just the opposite.
Some copyright law scholars argue that the digital age requires new userprivileges that grant users dramatically increased access to protected works
compared to the fair-use doctrine (Parchomovsky and Weiser, 2010). A policy
implication of this study would be the following: if a policy change that narrows
copyright protection is combined with copyright term extension, as currently
proposed by TPP negotiators, it would result in the least desirable outcome forboth social welfare and cultural prosperity.

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