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IOWA STATE UNIVERSITY
Department of Economics
Working Papers Series
Ames, Iowa 50011
Iowa State University does not discriminate on the basis of race, color, age, national origin, sexual orientation, sex, marital status, disability or status as a U.S.
Vietnam Era Veteran. Any persons having inquiries concerning this may contact the Director of Equal Opportunity and Diversity, 3680 Beardshear Hall, 515-
294-7612.
Welfare Effects of Intellectual Property Rights Under
Asymmetric Spillovers
Jeong Eon Kim, Harvey E. Lapan
October 2004
Working Paper # 04026
WELFARE EFFECTS OF INTELLECTUAL PROPERTY RIGHTS UNDER
ASYMMETRIC SPILLOVERS
Authors: Jeong-Eon Kim, Korea Institute of S&T Evaluation and Planning;
Harvey Lapan∗, Iowa State University
Abstract: We develop a model with one innovating northern firm and several heterogeneous
Southern firms that compete in a final product market. We assume the southern firms differ in
their ability to adapt technology and use this heterogeneity to study the differing incentives of
southern governments to protect intellectual property rights. We find that governments
representing more efficient firms have greater incentive to protect IPR than do those representing
less efficient firms. However, efficiency considerations imply that, given policies resulting in
the same overall innovation rate, it would be better to have weaker IPR protection for the more
efficient southern firms.
Classification Code: F13, O34
Key Words: Commercial Policy; Intellectual Property Rights protection; Trade; Innovation;
Imperfect competition
∗ Address of Corresponding Author: Harvey Lapan, 283 Heady Hall, Department of Economics, Iowa State
University, Ames, IA 50011; Tel: 515-294-5917; Fax: 515-294-0221; email: hlapan@iastate.edu
1
1. Introduction
The Uruguay round established a global agreement on intellectual property, which is
called TRIPS (Traded-related aspects of intellectual property rights). Under this agreement, most
developing countries should introduce the international minimum standards of protection by
2006. The recent debate in the WTO (World Trade Organization) meeting has been whether it is
desirable to extend IPR protection to the least developed countries. The declaration in the Doha
round extends the deadline for the least developed countries to introduce patent protection on
pharmaceuticals until 2016. This proposal seems reasonable since the least developed countries
do not have the capacity to absorb new knowledge from the innovations while they desperately
need the products developed by northern firms.
A number of papers deal with the issue of IPR protection in terms of North-South trade.
Chin and Grossman (1988) use a duopoly model to compare the welfare effects of IPR protection
between two regimes: ‘full IPR protection’ and ‘no IPR protection’. They show that the
economic interests of the North and the South are generally in conflict in the sense that ‘no IPR
protection’ benefits the South while it hurts the North. Diwan and Rodrik (1991) argue northern
and southern countries generally have different preferences for technology. They model the
‘appropriate technology’ for southern countries, and suggest that southern countries benefit from
IPR protection. Deardorff (1992) argues that, when IPR protection increases, the North is always
benefited while the South is hurt, and emphasizes that the effect on world welfare will be
negative if IPR protection is extended to all southern countries. Helpman (1993) suggests that
tightening IPR protection hurts both North and South in the presence of slow imitation while it
benefits only the North when the imitation rate is high. He also points out that higher protection
2
of IPR by the South could lead to slow innovation of northern firms, partly because of the lack of
competition.
Žigić (1998) extends Chin and Grossman’s model by introducing technological
spillovers to examine the role of IPR protection when only the northern firm conducts innovative
activity. The degree of spillovers is interpreted as an indicator of the inverse strength of IPR
protection. He shows that the South may benefit from tightening IPR protection through the
spillover effect of the increased northern firm’s R&D investment; however, by considering only
one Southern firm he effectively assumes all southern countries will have the same spillover rate.
Yang (1998) shows, using a partial equilibrium model, that both the North and the South would
be better off if some southern countries impose more IPR protection while the others impose less.
However, he does not identify which southern countries should provide more IPR protection for
the northern technology.1
By considering only one southern country and a common spillover parameter, Zigic
ignores the fact that the southern countries may face different spillovers. In Levin et al. (1987)
and Cohen and Levinthal (1989), firms may be different in their abilities to absorb or assimilate
intra-industry spillovers.2 We extend Žigić by introducing different spillovers among southern
countries to examine welfare effects of IPR protection. Only the northern country innovates, and
n-1 southern countries have different capacities to absorb knowledge spillovers from the northern
innovations.3 We assume, as in Žigić, the abilities to absorb spillovers in any southern country
decrease (increase) when IPR protection is tightened (relaxed). A two-stage game is considered.
1 This is because he assumes that all southern countries are identical.
2 Cohen and Levinthal (1989) calls this ability ‘absorptive capacity’.
3 In terms of the North, the issue of IPR protection may be ‘imitation’ of southern countries rather than spillovers.
Usually, ‘imitation’ and ‘spillovers’ are interpreted differently in the sense that ‘imitation’ is costly while
‘spillovers’ are costless. By different capacity to absorb spillovers, however, we are implicitly considering costly
spillovers. Thus, the terms ‘imitation’ and ‘spillovers’ are interchangeable in this paper even though we prefer
‘spillovers’, following Žigić.
3
In the first stage, the northern firm invests in R&D to create the new process. The outcome of
innovations reduces the unit production cost of the northern firm. The technology developed by
the northern firm provides benefits to the southern firms through spillovers. The degree of
spillovers is different across southern firms, depending on their ability to realize knowledge
spillovers. In the second stage, all firms engage in Cournot competition.
In this paper, we investigate the welfare effects of spillovers (or IPR protection), and
discuss the conflicts between the North and the South. The global welfare effects of spillovers
are also examined. Further, from the analysis, we identify which southern countries could be
benefited through tightening IPR protection. We find that more efficient Southern countries have
less incentive to increase spillovers than do less efficient Southern countries. But, from a world
welfare perspective, it is better that the more efficient countries expand the spillover. This
implies that private and social incentives may not be coordinated.
This paper is organized as follows. Section 2 presents the model and identifies the
equilibrium while section 3 provides comparative static analysis. Section 4 investigates the
welfare effects of spillovers and suggests some implications. The last section provides
conclusions.
2. The Model and Solution
There exist n countries in the world market: one northern country (labeled by 1) and n-1
southern countries (labeled 2,3,.,n). Each country has only one firm. All innovations take place in
the northern country, which conducts R&D. Through a spillover effect, n-1 southern countries
can partly appropriate the knowledge generated by the northern country, depending on their
4
knowledge absorptive abilities and the IPR protection level. Both North and South have access to
an old technology to produce a good demanded in the world market.
The northern firm has the following unit production cost function, which is the one
originally used by Chin and Grossman (1988):
()
12
1
C
α
γχ
=− , where
α
describes pre-
innovation cost, and
γ
is a parameter denoting the R&D efficiency. The term,
()
12
γ
χ
, represents
the R&D production function, which exhibits diminishing returns to scale with respect to R&D
investment,
χ
.4 The isouthern firm’s unit cost function is
th
()
12
ii
C
α
γχ
β
=− ,
where
ni ,....,3,2=
)1,0(∈
β
i denotes the index of spillovers or the strength of inverse IPR protection as in
Žigić (1998). The spillover parameter may consist of two terms: the IPR protection level and a
country-specific learning characteristic. The country-specific characteristic may include the
country’s ability to absorb R&D knowledge,5 or it may reflect imitation ability. Thus, even if
southern countries adopt a common IPR protection level, the value of the spillover parameter
may differ across southern countries, depending on their ability to absorb R&D knowledge.
Without loss of generality, we order the countries such that:6 . We assume
away two extreme cases,
βββ
n
>>> .......
32
0≡
β
i and 1
≡
β
i, which may reflect ‘perfect protection’ and ‘no
protection’ of intellectual property right, respectively.7
4 For more detail, see D’Aspremont and Jacquemin (1988) and Kamien et al. (1992).
5 Following Cohen and Levinthal (1989), we may call this ability ‘absorptive capacity’.
6 The amount of spillovers could be a choice variable, depending upon firm expenditures as well as local IPR
protection policy. However, throughout this paper we assume spillover expansion in any southern country increases
only when IPR protection is relaxed.
7 We could think of the spillover parameter as depending on (inverse) IPR protection, i
ρ
, and the country’s ability
to absorb knowledge
()
i
ω
. If
ω
ρ
β
i
ii += , then even if IPR protection is perfect ( 0=
ρ
i), spillovers may
occur. The results obtained in this paper do not hold if 0
≡
β
i.
5
Note that 1
1≡
β
in our set-up. By construction, the sum of spillovers is less than the
number of countries, i.e., . Consumers are assumed identical, and country i’s consumers
consume
∑<
=
n
jjn
1
β
[0,1]
i
θ
∈ proportion of total demand, which is given as a linear inverse demand
function: , .
QAP −= q
Qi
∑
=
The game among n countries consists of two stages, and we use the subgame perfect
Nash equilibrium. In the first stage, the northern firm chooses R&D investment, χ. In the second
stage, given the northern firm’s R&D investment, the n firms engage in Cournot-Nash
competition. To find the subgame perfect equilibrium, we first solve for the Nash equilibrium in
the second stage and then work backwards to solve for the first stage R&D level.
In the second stage, each firm maximizes its profit, which is given as:
()
()
() ii
ii
ii
PQ A Q c q
qq
c
π
= −=−− 1,...,in
∀
=
1
n
i
i
Q
=
≡q
∑
(1)
The first order condition for each firm (country) is:
()( )
0, 1,...,
ii ii
ddq Pcq i
π
=−−= = n (2)
Summing (2) across all firms, and assuming an interior solution for each firm yields:
()() ()
12
11
01 ;
NN
ii
ii
NP c Q N P A Nc c c N N
i
i
α
βγχ
==
⎛⎞⎛ ⎞
−−=→+=+ ≡ =−
⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
∑∑∑
(3)
Note that the solution has the well-known property (for linear systems) that the aggregate
equilibrium price and quantity depend upon the number of firms and average cost per firm, but
not on the distribution of the cost vector
(
)
1,..., n
β
β
. Using (2) and (3) yields:
6
()
()
()
()
()()
()
12 12
2
****
12
*
1
1;;
11
where
1
TT
i
iii
Tn
Tj
An nA
qqQ
nn
An
Pn
n
αββγχ αβγχ
π
αβγχ ββ
−+ + − −+
===
++
+−
==<
+∑
(4)
where the “*” indicates the equilibrium value.
In the first stage, given the second stage outcome, the northern firm chooses χ to
maximize its profit (including R&D cost):
()
**
** * * * *
11
111 11 1 1
1
;1
i
i
dq
dV dq dc
V P c q P c Pq Pq q
dddd
χχχχ
≠
⎛⎞
⎡⎤
⎛⎞
⎛⎞
⎛⎞ ⎛⎞
′′
⎡⎤
=− − =−+ + − +=
⎜⎟
⎜⎟⎢ ⎥
⎜⎟
⎜⎟ ⎜⎟
⎣⎦
⎝⎠ ⎝⎠
⎝⎠ ⎝⎠
⎣⎦
⎝⎠
∑1
0
χ
)
(5)
By assumption, , and the first term on the RHS of (5) is zero by the FOC for firm (1).
The last term on the RHS of (5) reflects the impact of R&D expenditures on firm 1’s total costs
whereas the middle term represents the strategic aspect of the firm’s decision, which arises only
because R&D decisions are made before output decisions. From (4) it is readily seen that:
(
1P′=−
() ()()
()
()
()
()
()
12 12
*1
1
11 21
21 21
TT
i
i
i
i
nn n
dq
dn n
ββγχ
βγ
χ
≠
≠
⎛⎞
+−−
⎜⎟
−+
⎛⎞
⎝⎠
==
⎜⎟ ++
⎝⎠
∑
∑
χ
(6)
The sign of (6) depends on T
β
, and hence the strategic interaction can increase or decrease firm
1’s investment in R&D. If
()
(
)
12
Tn
β
>+ , this interaction reduces the firm’s investment in
R&D, meaning that further R&D investment by firm 1 would lower its total costs but also lower
its profits due to the output effect on other firms.8 Using (4) in (5) and simplifying yields:
()
()
()
(
)
()
12
*
1
1
110
1
T
qn
dV d n
βγχ
χ
+−
=
+−=
(7)
It is readily seen that the second order condition holds. Solving (7), using (4), yields:
7
()
22
*
2
A
D
γα
χ
−∆
≡ where ;
11
T
n
β
∆≡ + − >
()
22
1Dn
γ
⎡
⎤
≡
+−∆
⎣
⎦ (8)
A meaningful (finite) solution to (8) requires
()
22
1n
γ
⎡0
⎤
+
−∆ >
⎣⎦
, which we assume
holds. Using equilibrium R&D in (4) yields equilibrium output levels and price:
()()
()
(
)
()
()()()
()
()()
()
()
12
*
*
*
111;
1
1
T
ii
i
T
An An
qn
Ann n
QD
αββγχ αγ
αγβ
−+ + − −+−∆−
==
+
−+−∆−
=
D
β
(9)
The equilibrium R&D level, and hence the aggregate equilibrium, depends on the
aggregate spillover
(
)
T
β
, but not on the distribution of spillovers among firms9. For all
countries to produce positive amounts, ∆−
+
≡< )1(
)1(
β
γ
γ
n
nn is required, which is equivalent to the
condition for the least productive southern country n to produce10.
It is desirable to compare the condition for the n-firm oligopoly equilibrium to exist in
our model with that for the duopoly equilibrium to exist both in Chin and Grossman (1988) and
in Žigić (1998). The conditions for the duopoly to exist in Chin and Grossman and in Žigić are
2/3<
γ
and )}2)(1{(3
β
β
γ
−−< , respectively. Two countries, the North and the South, are
modeled in both papers. Chin and Grossman consider ‘perfect protection’ of intellectual property
right while Žigić assumes that the southern country can take advantage of the benefits from
8Naturally, firm 1’s R&D investment is reduced by the presence of other firms. The strategic term merely shows
how R&D investment is affected by the fact it is chosen before outputs, rather than simultaneously with outputs.
9 If marginal costs – without R&D – differed across firms
(
)
ij
α
α
≠, the equilibrium level of R&D and thus the
aggregate equilibrium would depend on firm 1’s cost and average marginal cost, but not the vector of marginal costs.
8
northern firm’s innovation through spillovers. The condition in Chin and Grossman can be
recovered in our set-up by setting n=2 and 0
2
=
β
while Žigić’s condition is obtained by putting
n=2 and
β
β
=
2.
Both Chin and Grossman and Žigić consider two more types of equilibria: monopoly
and strategic predation. They show that the northern firm will enjoy the pure monopoly position
for a sufficiently high value of R&D efficiency parameter (
γ
) while it will act strategically to
induce southern firm’s exit (strategic predation) for an intermediate value of R&D efficiency.
These two types of equilibria can exist when there is more than one southern country in the
world. The monopoly condition11 in our set-up is
β
γ
2
1
2
−
>. The same condition for the
monopoly is obtained in Žigić where only one southern country is assumed. Note 2>
γ
is the
condition for the monopoly in Chin and Grossman where they consider perfect protection of
intellectual property rights ( 0=
β
). The condition for strategic predation is 22/3 <<
γ
and )1(2)}2)(1{(3
β
γ
β
β
−<<−− in Chin and Grossman and Žigić, respectively. In our set-
up, )1/(2)}1)(1{()1( 2
1
2
β
γ
β
β
−<<
∑
−+−+ =
n
jj
nn is the condition for strategic predation,
which is exactly the same condition as in Žigić if we assume that there exists only one southern
country in the market.12 Even though the outcome comparison among these equilibria is an
interesting issue, we do not consider these two equilibria since we are interested in investigating
the own and cross welfare effects of spillovers in the southern country.
()
()
()
22
1(1 ) 1
n
n
nn
γβ
⎡⎤
⎡⎤
≡+ −∆<+∆
⎣⎦
⎣⎦
(
)
(
)
1
T
nn
ββ
<+
()
12
n
β
<
10 Note if . Note that suffices
for this condition to hold.
11 This happens when ‘drastic innovation’ takes place, that is, for ni ,......,2
=
, where m denotes
monopoly outcome. Substituting price and R&D with monopoly outcome yields the condition.
)(
χ
m
i
mC
p<
12 The equilibrium R&D investment for strategic predation can be obtained by setting for . 0=
qi ni ,......,2=
9
3. Comparative statics
A change in the “spillover” rate in any southern country has direct and indirect effects.
The direct impact lowers costs in that country only, improving its competitiveness versus all
other countries (thereby hurting firms in those other countries). Since the increased spillover
lowers the private return to R&D, it causes the Northern firm to reduce R&D expenditures; this,
in turn, raises costs for all firms but raises costs most for those firms with large spillover rates.
Thus, the increased spillover in one country will likely harm not only the Northern firm but also
other firms with high spillover rates but may (will) benefit firms with very low spillover rates.
Proposition 1. An increased spillover rate in Southern country i reduces the Northern
firm’s R&D, raises costs for all other firms, but results in lower costs for firm i if its spillover
rate is sufficiently low.
Proof. Differentiating (8) yields:
() ()
()
22
*
** *
22
1
22
220; 0
1
i
n
d
dD n
γ
χγ
χχ χψ
βψ
γ
⎛⎞
⎡⎤
∆+−∆
⎛⎞
∆
⎛⎞
⎛⎞ ⎛⎞
−⎣⎦
⎜
≡− =−<≡ >
⎜⎟
⎜⎟
⎜⎟ ⎜⎟ ⎜
∆⎡⎤
⎜⎟
++∆
⎝⎠ ⎝⎠
⎝⎠ ⎝⎠ ⎣⎦
⎝⎠
⎟
⎟
(10)
()
()
() ()
*
12 12 12
***
*
2
j j
ij j ij
ii
dC d
dd
β
χ
γχ δ β γχ γχ δ
βχβψ
⎧
⎫
⎛⎞ ⎛ ⎞ ⎛⎞
⎪
⎪
=− − = − +
⎨
⎬
⎜⎟ ⎜ ⎟ ⎜⎟
⎪
⎪
⎝⎠
⎝⎠ ⎝ ⎠ ⎩⎭
(11)
where 1
ij
δ
= if i=j and zero otherwise. Since R&D falls, the costs of all firms - except firm i -
must increase. For j=i, then if i
β
is small enough costs fall. Formally:
0as
ii
i
dC
d
β
ψ
β
>>
⎛⎞
≤
⎜⎟
⎝⎠ ≤
. QED (12)
10
By construction, ; thus, fo
1∆> r
γ
sufficiently small, the inequality
(
)
i
β
ψ
< in (12) will be
satisfied for all Southern countries. Further,
ψ
is a decreasing function of
γ
, s
tant to note that the impact of increased spillovers on R&D depends only on the
o that the larger
the R&D efficacy, the less likely it is that a Southern firm will reduce its costs by increasing its
spillover.
It is impor
aggregate level of spillovers, and not which country is increasing its spillover. On the other hand,
the feedback effect of reduced R&D affects the high spillover firms more so that “low spillover”
firms have an incentive to increase their spillovers more than do high spillover firms.
Since the value of
ψ
recurs below, it is worthwhile simplifying the expression. Define:
()
()
()
()
() ()
()()
()
2
**
11 1 1n
βγβ
⎛⎞
*
2
*
11;
11
Tnnn
ββ ψγ γβ
⎡
⎤
+− −−
⎢
⎥
⎜⎟
⎣
⎦
≡+<+ ≡
⎜⎟
⎡⎤
+−
⎜⎟
⎢⎥
⎣⎦
⎝⎠
(13)
Turning to the impact of increased spillovers on that firm’s output, it is clear that its
output w s
for
ill increase if its unit production costs fall. However, since the costs of all other firm
must increase, it is possible for a firm’s output to increase even if its production costs rise;
clearly, what matters is how much its costs increase compared to the aggregate cost increase
all firms. Similarly, it is possible that the output of a “low-spillover” firm will increase, even
though it has not increased its own spillover rate. By the same logic, aggregate output could
increase if both the productivity of R&D investment and the aggregate spillover rate are low.
Formally, from (4):
()
()
()
()
()
(
)
12 1
11
1
Tj
j
ij
i
n
n
dn
ββ
δ
βψ
⎡⎤
⎛⎞
−+
⎢⎥
⎜⎟
=⋅+−+
⎜⎟
⎜⎟ ⎜⎟
+⎢⎥
⎝⎠ ⎝⎠
⎣⎦
(14)
*
dq
γχ
⎛⎞
11
()
()
()
()
()
(
)
()()
2
**
12 12
*
2
**
12 1
1
11
111
T
ii
dQ dP
ddn n
βγ β
γχ γχ
β
ββ ψ βγβ
−−−
⎡⎤
⎛⎞⎛ ⎞ ⎛⎞
−
=− = ⋅ + = ⋅
⎢⎥
⎜⎟⎜ ⎟ ⎜⎟
++
⎡
⎤
⎝⎠
⎝⎠⎝ ⎠ ⎣⎦ −−−
⎢
⎥
⎣
⎦
(15)
Proposition 2.
i. The equilibrium output of the firm which increases its spillover rate increases if and
only if:
()
()
()
1
T
inn
βψβ
<+ +
ii. If the aggregate spillover rate is sufficiently high, then the equilibrium output of a low
spillover firm may increase as a result of some other firm increasing its spillover rate:
i.e.,
()
()
()
()
2
**
*
2
**
211
0111
T
j
j
i
dq as
dn
βγβ
β
β
ββγβ
<>
⎛⎞
−+ −
⎛⎞ ⎛⎞
⎜⎟
≥≤
⎜⎟ ⎜⎟
⎜⎟
⎜⎟
⎜⎟ ++−
⎝⎠
⎝⎠ ⎜⎟
⎝⎠
,
ji≠
iii. If the productivity of R&D investment is not too high, then for low aggregate
spillover rates an increase in the spillover rate leads to higher aggregate equilibrium
output; i.e.,
()
**
2
*
12
01
i
dQ as
d
β
γ
ββ
<>
⎛⎞
⎛⎞ −
⎜⎟
≥≤
⎜⎟ ⎜⎟
⎝⎠ −
⎝⎠
Note that the second order condition requires
(
)
2
*
1
γβ
−
<− ; hence, condition (iii) must
hold for small spillover rates. This implies that over some interval higher spillover rates benefit
consumers as well as some firms. Note that for high aggregate spillover rates
()
*12
β
>, then
12
further increases in spillover rates must lower aggregate output.
4. Welfare effects
In this section we investigate the effect of a change in spillovers (or IPR protection) on
global welfare and welfare for each country. Since, from a global perspective, the original
equilibrium is inefficient, an increase in some spillover rate can have an ambiguous impact on
welfare. The inefficiency of the original equilibrium arises from several sources including: (i)
given the level of R&D, too little information is shared among countries; (ii) there is
underinvestment in R&D; (iii) given costs, too little output is produced; and finally (iv) the given
level of output is produced inefficiently since - under constant costs - all output should be
produced in the low cost country. An increase in the spillover rate to some country reduces the
inefficiency due to (i), exacerbates the inefficiency due to (ii); and - as seen in the previous
section - has an ambiguous impact on total output (and hence on the inefficiency due to (iii)).
The welfare of each (Southern) country consists of its firm’s (oligopoly) profits and
consumer surplus. Thus, for all countries but the Northern country, welfare is given by:
(16)
()
*;
jjj j j
j
WCSPCqCS
πθ θ
=+ =− + ≠1j
where CS is aggregate consumer surplus,
j
θ
is country j’s consumer share, and hence is
consumer surplus in country j. Differentiate (16) with respect to
jCS
θ
i
β
to obtain:
()
*
*;
jj j j
j
j
j
iii i ii i
dq
dW d dCS dP dC dP
PC q D j
ddd d dd d
π
βββ β ββ β
⎡⎤
⎛⎞
=+ =− + − − ≠
⎢⎥
⎜⎟
⎢⎥
⎝⎠
⎣⎦
1
j
(17)
Rearranging terms in (17) yields:
() ()
*
*
;
jj
j
jjj
j
iii i
dq
dW dP dC *
;1
j
j
X
qPC XqD
ddd d
βββ β
⎡⎤
⎛⎞⎛⎞
=−+− ≡−
⎢⎥
⎜⎟⎜⎟
⎢⎥
⎝⎠⎝⎠
⎣⎦
j
≠
(18)
13
where j
D
is consumption in country j. The first term in (18) represents the standard terms of
trade effect: an increase in world price benefits (hurts) a country if it is net exporter (net
importer). The second term is the benefit (cost) to the country, given output, due to the
exogenous change in unit production costs, while the third term reflects the change in monopoly
profits - at given price -due to the change in the firm’s output level. The firm’s profit
maximizing conditions imply:
*
*0;
j
j
jj
ii
dq dP dC
PC q ddd
i
β
ββ
⎛⎞
−−= → = −
⎜
⎝⎠
⎟
(19)
Substituting (19) into (18) and rearranging yields:
**
22 ;
jjj
j j
jj
iiii iii
dW dP dP dC dP dC dP 1
X
qqD
dddd ddd
ββββ βββ
⎡⎤⎡
⎛⎞⎛ ⎞ ⎛ ⎞
=+−=−−
⎢⎥⎢
⎜⎟⎜ ⎟ ⎜ ⎟
⎝⎠⎝ ⎠ ⎝ ⎠
⎣⎦⎣ j
⎤
≠
⎥
⎦
(20)
where
(
i
dP d
)
β
is given by (15) and
(
)
ji
dC d
β
by (11). Since an increase in the spillover rate
in some country i will lead to increased unit production costs in all other countries, it
immediately follows from (20) that:
Proposition 3: An increase in the spillover rate in Southern country i that leads to a lower world
price will:
(1)lower welfare in all other Southern countries that are net exporters.
(2)raise welfare in all countries that do not produce the good.
From Proposition 2, the increased spillover rate will lead to lower world prices when both the
aggregate spillover rate and the R&D productivity parameter are not too large. Lower world
prices, ceteris paribus, are bad for exporters and good for importers. Clearly, if
j
β
is
14
sufficiently low so domestic output is low, then the change in world price is the principal
determinant of the impact on domestic welfare. If world price increases, then Southern countries
with low output relative to demand must be hurt, whereas countries with no domestic demand
will be benefited if the price increase offsets the cost increase.
Clearly, how price changes relative to the firm’s cost is crucial in determining the welfare
impact of an increased spillover rate. From the linearity of the system we have, in general:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+−
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−→+=+ ∑∑ −
ββββ
i
j
ki
k
i
j
i
k
kd
dc
n
d
dc
n
d
dc
d
dP
c
APn )1(
)1(
)1( 1 (21)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛+−
+
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛−∑
−
ββββ
i
j
ki
k
i
j
id
dc
n
d
dc
n
d
dc
d
dP )1(2
)1(
21 (22)
For the firm’s costs will rise due to decreased R&D; thus, (21) and (22) imply that, if price
(average cost) increases, then countries with sufficiently low spillover rates will see their costs
rise less than price, and hence those countries will gain provided they are not big consumers of
the good. Using (15) and (11), we have, for the specific functional forms:
,ji≠
()
*
12 1
1
jj
ij
ii
dC
dP
dd n
ββ
γχ δ
ββ ψ
⎡⎤
⎛⎞
−
⎛⎞
−= ⋅−++
⎢⎥
⎜⎟
⎜⎟ ⎜⎟
+
⎝⎠
⎢⎥
⎝⎠
⎣⎦
;
()
()
()
()
()
2
*
1
2
**
11
11 11
n
γβ
ψ
γβ β
−
⎛⎞
+−
⎜⎟
≡⎜⎟
−− + −
⎜⎟
⎝⎠
(23)
()
*
12 2
1
22
1
j
ij
ii
dC
dP
dd n
ββ
γχ δ
ββ ψ
⎡⎤
⎛⎞
−
⎛⎞
−=⋅−++
⎢
⎜
⎜⎟ ⎜
+
⎝⎠
⎢⎥
⎝⎠
⎣⎦
j
⎥
⎟
⎟
(24)
Proposition 4: Suppose the spillover rate increases in Southern country i; then in other
Southern countries:
15
1. If net exports are zero, then domestic welfare will increase, remain unchanged or
decrease as:
()
()
()
2
**
2
*
21
21 1
j
βγ β
βγβ
>
⎛⎞
+− −
⎜⎟
≤⎜⎟
+−
⎜⎟
⎝⎠
1
2. If domestic demand is zero, then domestic welfare will increase, remain unchanged or
decrease as:
()
()
()
2
**
2
*
21
11
j
βγ β
βγβ
>
⎛⎞
+− −
⎜⎟
≤⎜⎟
+−
⎜⎟
⎝⎠
1
The proof follows immediately by substitution and rearrangement. Thus, if the countries are net
exporters, (only) low spillover countries can benefit from the increased spillover rate in some
other country. Note, however, that if price falls, the expression on the RHS of the inequality is
negative, and all net exporters must lose.
The country that increases its spillover rate is likely to benefit, since even though R&D
falls, the country appropriates more of the existing stock of knowledge. For a low spillover
country, it is fairly clear that it must benefit. Specifically:
Proposition 5: An increased spillover rate in country i will have the following impact on that
country:
1. If net exports are zero, then domestic welfare will increase, remain unchanged or
decrease as:
()()
()
()
()
()
()
2
** *
2
*
121 1 2 11 1
ˆ
21 1
ii
nn
βγ β β
ββ γβ
>
⎛⎞
+−−− +−−
⎜⎟
≤=
⎜⎟
+−
⎜⎟
⎝⎠
16
2. If domestic demand is zero, then domestic welfare will increase, remain unchanged or
decrease as:
()
()()
()
()
()
()
()
2
** *
2
*
111 1 11 1
11
ii
nn
βγ β β
ββ γβ
>
⎛⎞
+−−−− +−−
⎜⎟
≤=
⎜⎟
+−
⎜⎟
⎝⎠
The proof follows by substitution. Note that both ˆi
β
and i
β
are decreasing functions of
γ
; for
low R&D productivity, the country increasing its appropriation of foreign technology must gain.
However, if
γ
is large enough it is possible that the country could lose from doing so13; since the
SOC requires then a sufficient condition is:
()
2
*
1
γβ
−<1
Corollary:
1. If country i increases its spillover rate and its net exports are zero, then a sufficient
condition for its welfare to improve is:
(
)
*2
i
ββ
≤
2. If country i increases its spillover rate and its domestic consumption of this good is zero,
then a sufficient condition for its welfare to improve is: *
i
β
β
≤
Clearly, then, countries with current low spillover rates have an incentive to increase their
absorption of foreign knowledge (i.e., they have less incentive to strengthen IPR).
Next, consider the impact of the increased spillover rate on the Northern country.
Rewriting the profit function for the Northern firm:
()
()
11
PQ c q
1
π
χ
=−− (25)
Totally differentiating and using the FOC yields:
13 Of course, if the firm’s profits fall – which implies a decline in domestic welfare if there is no consumption, then
presumably a rational firm would not increase its spillover rate.
17
()
**
*
11 1
11
iiii
ddqdQdcd
Pc qP
ddddd
*
i
d
d
π
χχ
β
ββχβ
⎛⎞
⎛⎞ ⎛⎞
′
=− + − −
⎜⎟
⎜⎟ ⎜⎟
⎜⎟
⎝⎠ ⎝⎠
⎝⎠
β
where: (26)
**
**
1
1
1
jj
j
ii
qq
dQ dq
dd i
χ
β
ββχ
≠
⎛⎞
∂∂
∂
=+ +
⎜
⎜
∂∂∂
⎝⎠
∑
β
⎟
⎟
(27)
Substituting (27) into (26) and rearranging yields:
()
**
**
** * *
111
11 1 1 1
11
1jj
jj
ii ii
qq
ddqdcd
Pc q q q q
ddd d
πχ
*
1
j
j
i
q
β
βχ χβ β
≠≠
⎛⎞
⎛⎞ ⎛⎞ ⎛⎞
∂∂
⎛⎞ ⎛ ⎞
=−− − ++ − =−
⎜⎟
⎜⎟ ⎜⎟ ⎜
⎜⎟ ⎜ ⎟
⎜⎟ ⎜⎟ ⎜
⎜⎟
∂∂
⎝⎠ ⎝ ⎠
⎝⎠ ⎝⎠ ⎝⎠
⎝⎠
∑∑
β
≠
∂
⎟
⎟
∂
∑
(28)
where the first two terms on the RHS vanish due to the envelope theorem. The impact of an
increase in i
β
on firm 1’s profits is due to the increase in output of all Southern firms, given the
level of R&D. From (4),
()
() ()
(
)
12
*
1
21
ji
j
qn
βγχ
≠
∂∂= +>
∑0
so, not surprisingly, the profits of
the Northern firm must fall. The impact on welfare then hinges on how consumer surplus
changes; a resulting increase in world prices – due to the reduced R&D - must hurt the North,
while it might gain if prices fall. Formally:
()
()
()
()()
()
2
12 **
11
11
1
2
**
12 1 2
1111
iii
dW dP d
DD
dddn
βγ β
γχ
π
βββ βγβ
⎛⎞
⎧⎫
⎛⎞ −−−
⎜⎟
⎪
=− + = ⋅ −
⎜⎟
⎨
⎜⎟
⎜⎟
+−+−
⎪⎪
⎜⎟
⎝⎠
⎩⎭
⎝⎠
*
q
⎪
⎬
(29)
Proposition 6: A sufficient condition for increased spillovers to harm the North is
. If world price rises, the North must be hurt.
))1/()21(()2/( **
1
*
1
ββ
−−⋅≥ D
q
Proof: From (29), the term inside the
{
}
on the RHS is a decreasing function of
γ
. Thus, the
term inside the parentheses reaches a maximum at 0
γ
=
, from which the result follows. If
world price increases both terms on the RHS are negative and hence
(
)
10
i
dW d
β
<.
QED
18
Finally, consider the impact of increased spillovers on world welfare (the sum of
surpluses in all countries). This can be obtained by summing (17) over all Southern countries
and adding (29), or directly from the definition of welfare14:
(30)
()
()
11
0
,;
Qnn
Tjjj
jj
WPydy c q Q
χβ χ
==
⎛⎞
≡− −=
⎜⎟
⎝⎠
∑
∫j
q
∑
where stands for world (“total”) welfare. Differentiating (30) yields:
T
W
()
**
*
11
;
Tnn
jj
jj
jj
iii
dq dc
dW d
Pc q
ddd
χ
βββ
==
⎛⎞
≡− − −
⎜⎟
⎝⎠
∑∑i
d
β
(31)
The first term represents the surplus created from increased output since in the original
equilibrium price exceeded marginal cost while the second and third terms (collectively)
represent the surplus created by the net reduction in production costs. It is well known that in the
linear model total output, and hence price, depends only upon average marginal cost of the firm,
while total profits depends upon the variance of the cost vector. Hence, if all firms were alike, a
sufficient condition for total welfare to increase as a result of the increased spillover is that total
output did not fall, since expenditures on R&D fall and since total output moves in the opposite
direction from average cost. However, since the output vector of the firms matters, we are also
concerned with whether the proportion of output produced by the low cost firms rises or falls
(i.e., whether the variance of the cost vector rises or falls).
Formally, from (3) and (4):
() ()
12
11
;;; ;
11
nn
ii
ii
c
nA c
Anc Q
PQ qc
nn nn n
β
T
n
β
αβγχ β
==
−
⎛⎞
+
⎛⎞ ⎛⎞
== ≡≡=−≡
⎜⎟
⎜⎟ ⎜⎟
++
⎝⎠ ⎝⎠
⎝⎠
∑∑
=
(32)
Define:
output vector, not just total output. The reason for the former is that price is equal for all consumers, while the latter
14 Note that the consumption value depends only on total output (consumption) whereas the costs depend on the
19
()
()
12 ;
jj j jj
cc qq j
ε
ββγχ ε ε
⎡⎤
=− →=− =+
⎣⎦ where, by construction: (33)
1
0
n
j
j
ε
=
=
∑
Thus, (30) and (31) can be rewritten as:
()
(
)
(
)
22 2
222
Tj
jj
WAQQ cQ n nQ 2
j
ε
χ
=− −+ −=+ −+
∑
χε
∑
(34)
()
()
()
2
2
j
Tj
iii
d
Qd Q
dW d
nn
ddd
i
d
ε
χ
βββ
⎛⎞
⎜⎟
⎝
=+ − +
β
⎠
∑
(35)
Thus, (34) shows the role played by both average costs and the “variance” of these costs, while
(35) reaffirms the claim – if we could ignore the impact of this variance – that if output does not
decrease, total welfare must increase since we know
(
)
0
i
dd
χβ
<
. Using (33):
()
()
()
() (
2
22
22
1
1
22;
2
jn
j
ii
j
ii
dd
dd
β
ββ
εσ
χ
)
j
γχ β β
σ
γχ β β
σ
ββ
ββχψ
=
⎛⎞
⎜⎟ ⎛⎞
⎛⎞
⎛⎞
⎛⎞
⎛⎞
⎝⎠
=−+ =−− ≡−
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠
⎝⎠
⎝⎠
⎝⎠
∑∑ (36)
where 0
ψ
> is defined earlier. Thus, if i
β
β
≤
, the variance in the cost structure is reduced
(due to reduced R&D), and this tends to lower welfare, whereas for i
β
β
> the change on the
variance of cost is ambiguous. Turning to the first two terms in (35) we have:
()
(
)
() (
*
11
222; 1
1
T
ii
m
Qd Q
ndn m
nd d n m
γβ
χχ
)
ψ
ββ
ββψ
⎧⎫
⎛⎞
−−
⎛⎞
++
⎪⎪
⎛⎞ ⎛⎞
⎜⎟
−= −+ ≡−
⎨⎬
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
+
⎝⎠ ⎝⎠
⎝⎠
⎪⎪
⎝⎠
⎩⎭
(37)
While this expression can be further simplified, there is not much more learned by doing so.
Thus, we conclude:
Proposition 7: Given an increased spillover rate in one country, then:
applies because marginal cost is not equalized across firms.
20
1. Total welfare is more likely to increase when this increase in spillover occurs in a country
which already has a high spillover rate.
2. Total welfare will increase when aggregate output increases, provided the dispersion in
spillover rates is not too large.
5. Conclusions
This paper has investigated welfare effects of spillovers due to relaxed IPR protection.
Unlike previous studies where two countries, North and South, are modeled, we consider the
situation where there exist many southern countries in the market. One important feature in the
model is to distinguish southern countries according to the absorptive capacity to realize
spillovers. This is crucial in analyzing the conflicts among southern countries on the issue of IPR
protection. A number of results are obtained from the analysis. An increased spillover rate in
Southern country i reduces the Northern firm’s R&D, raises costs for all other firms, but results
in lower costs for firm i if its spillover rate is sufficiently low. It is clear that a firm’s output will
increase if its unit production costs decrease with spillovers. However, since the costs of all other
firms must increase, it is possible for a firm’s output to increase even if its production costs rise;
what matters is how much its costs increase compared to the aggregate cost increase for all firms.
Similarly, the output of a “low-spillover” firm may increase as a result of some other firm
increasing its spillover rate.
The welfare of each Southern country consists of its firm’s profits and consumer surplus.
When an increase in the spillover rate leads to a lower world price, Southern countries with high
output relative to demand (net exporters) are hurt, whereas southern countries with no domestic
demand are benefited. On the other hand, when world price increases with the spillover
21
expansion in a southern country, Southern countries with low output relative to demand must be
hurt, whereas countries with no domestic demand will be benefited if the price increase offsets
the cost increase. The Northern firm is always hurt whenever the spillover rate in a Southern
country increases because it results in the decline of Northern firm’s output, but the increase in
collective output of all Southern countries. A resulting increase of world price, therefore, must
hurt the North, while it might gain if prices fall.
Note that more efficient Southern countries have less incentive to increase the spillover
than do less efficient countries. The reason is, as shown above, the feedback effect of reduced
R&D impacts the high spillover firms more than low spillover firms. But, from a world welfare
perspective, it is better that the more efficient countries expand their spillover. This implies that
private and social incentives may not be coordinated. One thing to note is that the one-size fits all
agreement is probably not optimal because the ability to absorb the Northern knowledge is
different among Southern countries.
There are some possible extensions of this study. How much each country absorbs the
knowledge or information from another country depends on its ability to realize knowledge
spillovers. Thus, it will be interesting to introduce endogenous spillovers by having a cost
function: ),(
α
µ
β
i
ii where
µ
i is the cost of reverse-engineering and
α
i is a country-specific
parameter. Given a vector of
α
, we could model the “spillover” decision without IPR and then
have IPR shift the cost function. Second, the existence of spillovers may increase the northern
firm’s incentive to sell its innovations to the southern countries. Thus, the issue of licensing may
be an important topic for future research. Third, the direct extension of this paper would be to
investigate optimal patent policy in terms of domestic welfare or how to reach an agreement on
IPR protection that is Pareto improving.
22
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