Technology transfer in a linear city with non symmetric locations

Technology transfer in a linear city with non
symmetric locations
Fehmi Bouguezzi 1
LEGI and Faculty of Management and Economic Sciences of Tunis
Abstract
This paper compares patent licensing regimes in a Hotelling model where we can
…nd two …rms located respectively in the …rst half of the city and in the second half.
I suppose that one of the …rms has a process innovation reducing the marginal unit
cost. This patent holding …rm will decide to license or not the non innovative …rm
and will choose, when licensing, between a …xed fee or a royalty. The key di¤erence
between this paper and other papers is that here I suppose that …rms can change
their locations on the linear city. I show that when there is no licensing, the non
innovative …rm makes non positive pro…t when innovation is drastic. I also show
that choice between a …xed fee or a royalty depends on the size of innovation and
locations of innovative and non innovative …rms. Finally, I …nd that no licensing is
the best strategy for the patent holding …rm when innovation is drastic.
Key words: Keywords: Hotelling model, Technology transfer, Patent licensing
JEL Codes: C21, L24, O31, O32
1
Introduction
Several authors studied patent licensing and transfer of innovation. Wang
(1998) and (2002) compared licensing regimes in a Cournot duopoly and then
in a di¤erentiated Cournot oligopoly and …nd that optimal licensing regime
depends on the size of the innovation (drastic or non drastic). Kamien, Oren
and Tauman (1992) studied optimal licensing regime for a cost reducing innovation when innovative …rm is outside of the market. Cohen and Morrison
1
Email : [email protected]. Address: Loboratory of Management and
Industrial Economics , Polytechnic School of Tunisia. El Khawarezmi Rd B.P. 743,
2078 La Marsa - Tunisia. Phone. (+216)71774611 - Fax (+216)71748843
1
(2004) focused on spillovers in the US food manufacturing industry across
states and from agricultural input supply and consumer demand and …nd average and marginal cost e¤ects in the spatial and industry dimensions that
a¤ect location decisions. Mai and Peng (1999) discussed cooperation and competition between …rms in a Hotelling spatial model with di¤erentiation. Piga
and Theoloky (2005) supposed that R&D spillovers depend on …rm’s location
which means that spillovers increase when …rms are close the each others.
They show that distance between …rm’s location increases with the degree of
product di¤erentiation. Osborne and Pitchik (1987) studied optimal locations
of two competing …rms in a Hotelling’s model and …nd that they choose locations close to the quartiles of the market. Paci and Usai (2000) investigate
the process of spatial agglomeration of innovation and production activities
in an econometric analysis of 85 industrial sectors and 784 Italian local labor
systems and …nd that technological activities of a local industry in‡uence positively innovations of the same sectors in contiguous areas. Alcacer and Chung
(2007) examine …rms’location choices expecting di¤erences in …rm’s strategies
of new entrants into the United States from 1985 to 1994 and …nd that …rms
favor locations with academic innovation activity. Alderighi and Piga (2009)
investigate properties of two types of cost reducing restrictions that guarantee
the existence of equilibrium in pure strategies in Bayesien spatial models with
heterogeneous …rms. Poddar and Sinha (2004) studied technology transfer in
a Hotelling model where …rms are located at the end points of the linear city
and …nd, for an insider patentee, that royalty licensing is optimal when innovation is non drastic while no licensing is the best when innovation is drastic.
Matsumura and Matsushima (2008) studied the relationship between licensing
activities and the locations of the …rms. They …nd that licensing activities following R&D investment always lead to the maximum di¤erentiation between
…rms and the mitigation of price competition. Long and Sonbeyran (1998)
supposed in a Hotelling model that spillovers depend on the distance between
…rms and …nd that agglomeration can be optimal. They also …nd that geographical dispersion in a two dimensional plane is another possible outcome.
Hussler, Lorentz and Rond (2007) supposed that, in a Hotelling model, absorptions capacities of the …rms are function of their internal R&D investment
and …rms determine endogenously the maximum level of knowledge spillovers
they might absorb. They …nd that knowledge spillovers are maximum if …rms
are located symmetrically and tend to agglomeration in the center of the linear city when transportation cost increase. Pinkse, Slade and Brett (2002)
investigate the nature of price competition among …rms producing di¤erentiated products and competing in markets that are limited in extent through an
econometric study of the US wholesale gasoline markets and …nd that competition is highly localized. Alderighi and Piga (2008) considered a Salop model
with heterogeneous costs and …nd that cost heterogeneity increases welfare
and induce less excessive entry. This paper tries to study optimal licensing
regimes when …rms are not located at the end points of the linear city (Poddar and Sinha 2004) but are dynamic. The major result is that royalty is not
2
always optimal when innovation is not drastic since non innovative …rm makes
a non positive pro…t when the size of innovation is small and so, we cannot
…nd any regime of licensing in this case. Innovative …rm will not license its
innovation and become a monopoly. Another result shows that no licensing is
better that …xed fee licensing independently of the size of innovation.
2
Model
Let’s suppose a linear city with a long l and two …rms A and B producing
homogeneous goods and located respectively on the …rst half and the second
half of the city. The …rm A is located at a ( 0 < a < 2l )and the …rm B at b
( 2l < b < l).
firme B
firme A
0
a
l
b
l
2
l
a ∈ [0, ]
2
l
b ∈[ , l]
2
To compare patent licensing regimes, I suppose that …rm A owns a patented
cost reducing innovation allowing to reduce the unit marginal cost by " which
measures the size of the innovation and depends on the investment on R&D
by innovative …rm. Consumers are uniformly distributed on the linear city
(interval [0; l])and each one pay a linear transport cost equal to td (t is the
transport unit cost and d the distance between the consumer and the …rm).
The innovative …rm will choose between two licensing regimes : a …xed fee
licensing where non innovative …rm must pay an amount of money not depending on the quantity produced in exchange of the use of the license or
a royalty licensing where non innovative …rm must pay a …xed rate on each
3
quantity produced using the new technology. Game stages are as follows: in
the …rst stage, the two …rms choose their locations. In the second stage, decides to license its innovation or not and the …xed fee or the royalty to apply
and in the third and last stage, the two …rms compete in prices. To calculate
demand functions of the two …rms, we must …nd the location of the marginal
consumer where its utility function when buying the product of the …rm A is
equal to its utility when buying the product of the …rm B: The utility of each
consumer depends negatively of the transportation cost and the price of the
product.
UA =
p1
t jx
aj and UA =
p2
t jx
bj
The utility function of a consumer located at x and buying the …rm A product
is :
8
>
<
UA = >
:
p1
t (a
x) if x < a
p1
t (x
a) if x > a
The utility function of a consumer located at x and buying the …rm B product
is :
UB =
0
8
>
<
>
:
p2
t (b
x) if x < b
p2
t (x
b) if x > b
)
x
a
b
l
− p1
− p2
UB
− p1 − t (b − a )
− p2 − t (b − a )
UA
4
In the interval x 2 [a; b] the location of the marginal consumer is x~ and veri…es
UA = UB () x~ = p2 p1 +t(a+b)
2t
Demand function of the …rm A is :
DA =
8
>
>
>
l
>
>
<
if p1 2
8
>
>
>
>
>
<
IntA
1
() DA =
if p1 2 IntA
2
>
>
>
>
>
A
:
if p1 2 Int3
x~
>
>
>
>
>
: 0
IntA
1 = [c1 ; p2
where IntA
2 = [p2
t(b
t(b
p2 p1 +t(a+b)
2t
if p1 2 IntA
1
if p1 2 IntA
2
if p1 2 IntA
3
0
a)]
a); p2 + t(b
IntA
3 = [p2 + t(b
l
a)]
a); +1[
Demand function of the …rm B is :
8
>
>
>
>
>
<
DB = > l
>
>
>
>
:
if p2 2
0
x~
l
8
>
>
>
>
>
<
IntB
1
() DB =
if p2 2 IntB
2
>
>
>
>
>
B
:
if p2 2 Int3
IntB
1 = [p1 + t(b
where IntB
2 = [p1
if p2 2 IntB
1
p1 p2 +t(2l a b)
2t
if p2 2 IntB
2
l
if p2 2 IntB
3
a); +1[
t(b
IntB
3 = [c2 ; p1
0
a); p1 + t(b
t(b
a)]
a)]
Pro…ts of the innovative and non innovative …rms are :
A
B
8
>
>
>
>
>
<
(p1
c1 ) l
(p1
>
>
>
>
>
:
c1 ) p2
= > (p2
c2 ) p1
=
8
>
>
>
>
>
<
>
>
>
>
:
p1 +t(a+b)
2t
if p1 2 IntA
1
if p1 2 IntA
2
if p1 2 IntA
3
0
0
(p2
p2 +t(2l a b)
2t
c2 ) l
if p2 2 IntB
1
if p2 2 IntB
2
if p2 2 IntB
3
To …nd a Nash equilibrium, the prices of the two …rms p1 and p2 must verify
this inequality : jp1 p2 j
t(b a). In fact, the pro…t of the …rm A is not
positive in the interval IN T3A and to make a positive pro…t, …rm A should
choose a price p1 verifying p1 < p2 + t(b a):Also, …rm B realize a non
positive pro…t in the interval IntB
1 and should choose a price p2 verifying
5
p2 < p1 + t(b a): We show …nally that a Nash equilibrium exists in the
B
interval IntA
2 (or Int2 ).
Pro…ts maximization in respect of prices gives:
8
>
<
>
:
@ A
@p1
=0
@2 A
@p21
and
<0
8
>
<
@ B
@p2
>
:
=0
@2 B
@p22
<0
We …nd at the equilibrium :
8
>
<
>
:
@ A
@p1
=0
@ B
@p2
=0
()
8
>
< p1 =
>
:p =
2
1
2
1
2
(p2 + t (a + b) + c1 )
(p1 + t(2l
a
=)
b) + c2 )
8
>
<p =
1
>
:p =
2
1
3
(t(2l + a + b) + 2c1 + c2 )
1
3
(t(4l
a
b) + c1 + 2c2 )
The optimal pro…t functions of …rms A and B at the equilibrium are :
A
=
1
18t
c1 + c2 )2 and
(t(2l + a + b)
=
B
1
18t
(t(4l
a
b)
c1 + c2 )2
Demand functions are :
DA = x~ =
DB = l
3
1
6t
(2tl + t(a + b)
x~ =
1
6t
(4tl
c1 + c2 ) if p1 2 IntA
2
c2 ) if p2 2 IntB
2
t(a + b) + c1
No licensing
In this regime, innovative …rm pro…t alone from its innovation while non innovative …rm uses the old technology. Denoting by c1 and c2 marginal unit
costs of respectively …rm A and …rm B, we can write: c1 = c " and c2 = c.
Replacing in …rms equilibrium pro…ts, we …nd:
PL
A
=
1
18t
(t(2l + a + b) + ")2 and
PL
B
=
1
18t
(t(4l
a
b)
")2
Price equilibrium are :
p1 = 31 (t(2l + a + b) + 3c
2") and p2 = 13 (t(4l
a
b) + 3c
")
Proposition 1 when …rms are located on the …rst and second half of the linear
city and when there is no licensing, the non innovative …rm leave the market
when innovation is drastic.
PROOF. p2 = 13 (t(4l
a
b) + 3c
"):
6
p2 > c if " < t(4l a
(drastic innovation)
b (non drastic innovation) and p2 < c if " > t(4l
a
b)
Proposition 2 When innovation is not drastic and not licensed, the two
competing …rms agglomerate, in the equilibrium, in the center of the city
a = b = 2l
PROOF. @@aA = 19 (t (2l + a + b) + ") > 0. Since …rm A is located in a with
0 a 2l then optimal location for …rm A is a = 2l :
@ B
@b
= 19 (t (4l a b) ") : Firm B optimal location depends on the size
of the innovation. In fact, when innovation is non drastic (" < t(4l a b))
b
l: However,
then @@bB < 0 and …rm B will locate in b = 2l since 2l
@ B
when innovation is drastic (" > t(4l a b)) then @b > 0 and b = l (in
this case …rm B leave the market since it don’t bene…t of a license of a drastic
innovation)
Equilibrium pro…t functions …r a non drastic innovation are :
PL
A
4
=
1
18t
(3tl + ")2 and
PL
B
=
1
18t
(3tl
")2
Fixed fee licensing
In this regime, …rm B can use the new technology in exchange of the payment
of a …xed fee denoted by F to the patent holding …rm. The maximum amount
that …rm A can choose is equal to the increase of …rm B pro…t when using
PL
with
! 0 to be sure that …rm B
the new technology. F = FB
B
will accept to buy the license.
Firm A and …rm B production unit costs are c1 = c2 = c ". Replacing in
the pro…t functions we …nd : FA = 18t (2l + a + b)2 and FB = 18t (4l a b)2
Fixed fee amount is equal to :
F =
"
9t
4tl
t (a + b)
"
2
Total revenue of the patent holding …rm is:
F
A
=
F
A
+F =
t
18
(2l + a + b)2 +
"
9t
4tl
t (a + b)
Firm B total pro…t after the payment of royalties is :
7
"
2
F
F
B
=
B
F =
t
18
(t(4l
a
")2
b)
Lemma 3 In a …xed fee licensing, non innovative and patent holding …rm
agglomerate in the center of the city when innovation is strong but non drastic.
For a small or intermediate innovation, …rms are dispersed.
PROOF.
@ F
A
@a
1
9
=
("
t(2l + a + b)) :
@ F
A
@a
@
F
> 0 if " < t(2l + a + b) and @aA < 0 if " > t(2l + a + b):Since a 2 [0; 2l ]
then a = 0 if " < t(2l + a + b) and a = 2l if " > t(2l + a + b):
@
F
= 9t (" t(4l a b)) < 0 because we are working on the non drastic
innovation case which means that " < t(4l a b). Since b 2 [ 2l ; l] then b = 2l
B
@b
Firm A total revenue can be written as follows:
F
A
8
>
< 1 tl2 + 1 "l
2
3
=>
:
25 2
tl
72
+
"2
18t
if 0 < " < t(2l + a + b)
"2
18t
7
"l
18
if t(2l + a + b) < " < t(4l
a
b)
0 < " < t(2l + a + b)
PL
A
PL
F
A
F
=
1
18t
(3tl + ")2 ;
"2
18t
= 21 tl2 + 13 "l
PL
B
;
=
F
1
18t
1
18t
=
B
t(2l + a + b) < " < t(4l
PL
F
A
F
=
(3tl + ")2 ;
PL
A
=
1
18t
25 2
tl
72
+
7
"l
18
"2
18t
PL
B
;
=
F
B
")2 +
(3tl
a
1
18t
=
")2
(3tl
l
2
a = 2l , b =
l
2
b)
(3tl
1
18t
a = 2l , b =
7
tl
2
")2
"
2
+
a = 2l , b =
l
2
a = 0, b =
l
2
Proposition 4 No licensing is always better than …xed fee licensing for a
patent holding …rm on a Hotelling model where …rms share equally their locations on the linear city.
PROOF.
F
A
PL
A
=
(since here l
F
A
PL
A
=
11 2
1
tl + 18
"
72
"
2 t < 0)
"2
9t
l
< 0 if 0 < " < t(2l + a + b)
2 "t
< 0 if t(2l + a + b) < " < t(4l
8
a
b)
5
Royalty licensing
In the royalty regime, the cost-reducing innovation is sold to the non innovative
…rm in exchange of a royalty amount depending on the production made with
the use of the new technology. The amount of royalties is proportional to the
demand of …rm B and equal to r (l x~). Firm B will accept to buy the license
in this regime only when it will allow it to increase its no licensing pro…t which
means that r must be in the interval ]0; "[ unless this licensing regime will not
be important to study.
Production unit costs of …rms A and B are respectively c1 = c
c " + r. Replacing in the price functions we …nd :
p1 = 31 (t(2l + a + b) + 3c
3" + r) and p2 = 13 (t(4l
a
b) + 3c
" and c2 =
3" + 2r)
pro…t equilibrium functions are
r
A
=
1
18t
(t(2l + a + b) + r)2 and
r
B
=
1
18t
(t(4l
a
r)2
b)
To …nd optimal royalty rate for innovative …rm, we should …nd the total revenue of …rm A and then maximize it with respect to r. Firm A total revenue
is:
r
A
8
>
<
>
:
=
r
A
+ r(l
@ rA
=0
@r
@ 2 rA
=
@r2
2
9t
x~) =
1
18t
(t(2l + a + b) + r)2 +
t
4
=) r = 4tl
<0
r
6t
(4tl
t (a + b)
r)
(a + b)
Since r < " < t(4l a b) then r = 4tl 4t (a + b) > " and so r = "
(with ! 0). In fact, optimal per unit royalty must not exceed ". Since rA
is an increasing function of r on the interval [0; "] then the per unit royalty
maximizing rA is ": Firm B will accept to buy a license only when it makes
a pro…t higher than the non licensing regime which means that it will accept
this licensing regime when the per unit royalty is lower than ". We will take
the optima per unit royalty r = "
where close to 0:
Total revenues of …rm A and …rm B after licensing are :
r
A
r
B
=
=
1
18t
1
18t
(t(2l + a + b) + ")2 +
(t(4l
a
b)
")2
"
6t
(4tl
"
6t
(4tl
t (a + b)
")
t (a + b)
")
Lemma 5 In a royalty licensing, …rms agglomerate when innovation is small
( a = b = 2l if 0 < " < 58 tl 52 t(a + b)) and are dispersed for an intermediate
innovation (a = 2l and b = l if 85 tl 52 t(a + b) < " < 4tl t(a + b)).
9
@
r
1
PROOF. @aA = 18
(" (4tl + 2t (a + b))) < 0 because 4tl
4tl + 2t (a + b) . Since 0 a 2l then a = 2l
r
@
=
B
@a
r
@
l
2
B
@a
5
18
8
tl
5
"
2
t (a
5
< 0 if " < 85 tl
b l
t (a + b) <
+ b)
2
t(a
5
+ b) and
r
@
> 0 if " > 85 tl
B
@a
2
t(a
5
+ b) where
Total revenues of innovative and non innovative …rms after choosing their
locations can be written as :
0 < " < 85 tl
PL
A
PL
r
A
R
1
18t
=
=
(3tl + ")2 ;
1
18t
(3tl + ")2 +
"
6t
8
tl
5
PL
A
PL
r
A
R
=
1
18t
7
tl
2
2
+"
(3tl
2
t(a
5
=
+
2
t (a
5
1
18t
") ;
PL
B
r
B
=
1
18t
1
18t
(3tl
")2
(3tl
")2
+ b) < " < 4tl
(3tl + ")2 ;
"
6t
=
+ b)
5
tl
2
" ;
PL
B
r
B
(3tl
")
1
18t
1
18t
(3tl
5
tl
2
")2
8
tl
5
"
5
tl
6t 2
If
2
t(a
5
"
8
tl
5
2
t (a
5
+ b) < " < 4tl
1
18t
+ b) then
t(a + b) then
"
2
"
6t
5
tl
2
"
r
A
r
A
PL
= 6t" (3tl
A
1
7
PL
= 18t
tl
A
2
") > 0:
+"
2
+
2
(3tl + ") > 0
Lemma 7 For a small innovation, a Nash equilibrium do not exist for a royalty licensing and then we have not necessarily two competing …rms on the
linear city.
PROOF. See Appendix
6
Fixed fee versus royalties
Total revenues of patent holding …rm under …xed fee licensing and royalties
licensing calculated after their optimal positioning on the linear city and depending on the size of innovation are :
10
a =b =
l
2
a =b =
Proposition 6 Royalties licensing is better than …xed fee licensing for a patent
holding …rm when innovation is non drastic.
PROOF. If 0 < " <
l
2
t(a + b)
=
=
"
6t
a =b =
a =
l
2
l
2
,b =l
0 < " < t(2l + a + b)
F
A
F
F
A
F
"2
18t
= 21 tl2 + 31 "l
=
;
F
1
18t
=
B
t(2l + a + b) < " < t(4l
a
b)
"2
1
18t
7
tl
2
2
t (a
5
+ b)
r
1
18t
25 2
tl
72
+
7
"l
18
;
18t
F
B
=
0 < " < 85 tl
r
A
R
r
A
R
=
=
1
18t
(3tl + ")2 +
1
18t
")2 +
(3tl
7
tl
2
+"
8
tl
5
2
"
6t
(3tl
2
t(a
5
+
"
6t
") ;
B
=
"
(3tl
+ b) < " < 4tl
5
tl
2
" ;
r
B
=
2
a = 2l , b =
+
l
2
a = 0, b =
")2
"
6t
(3tl
")
l
2
a =b =
t(a + b)
1
18t
5
tl
2
"
2
"
6t
5
tl
2
"
a =
Lemma 8 Royalties licensing is better than …xed fee licensing when innovation is non drastic.
PROOF. If 0 < " < 58 tl
2
t (a
5
+ b) then
r
A
F
A
=
"
6t
("
3tl) > 0
1
2
F
t(a+b) < " < 2tl+t(a+b) then rA
"2 17
tl"
A =
5
18t
2
p
p
0 since "0 = 17 4 341 tl < 0 < 2tl + t(a + b) < "00 = 17+4 341 tl:If
1
F
tl" 6t2 l2
b) < " < 4tl t(a + b) then rA
"2 15
A =
18t
2
p
p
"0 = 15 4 321 tl < 0 < 4tl t(a + b) < "00 = 15+4 321 tl
If 85 tl
13 2 2
tl
4
>
2tl + t(a +
> 0 since
Proposition 9 Patent holding …rm license its innovation under royalties regime
only when innovation is intermediate. For a small innovation, it will bene…t
alone from the new technology and become a monopoly for a drastic innovation..
7
Conclusion
We studied in this paper a Hotelling model on a linear city where …rms are
located in …rst and the second halfs of the city. We supposed that one of them
has a patented cost reducing innovation. We studied two di¤erent licensing
contracts: a per unit royalty and a …xed fee and we found that optimal locations of the two …rms depend on the licensing regime and the size of the
innovation. We found that on a hotelling model, no licensing is always better
than …xed fee regime independently of innovation size (drastic and non drastic
innovations). We showed that for a non drastic innovation, royalty licensing
11
l
2
l
2
,b =l
is better than …xed fee licensing for the patent holding …rm. Results showed
also that royalty licensing is better than no licensing for non drastic innovation and is optimal only when innovation is intermediate. In fact, for a small
innovation, a Nash equilibrium do not exist and we can not discuss licensing
regimes with only one …rm on the city. Finally, when innovation is drastic,
innovative …rm will bene…t alone of its innovation and become a monopoly.
Appendix
Innovative …rm pro…t consists of three di¤erent functions : a¢ ne, parabolic
and null functions
8
>
>
>
>
>
<
>
>
>
>
>
:
A
= (p1
c1 ) l
If
r
A
= (p1
c1 ) x~
If
A
=0
p1 < p 2
p2
t(b
If
t(b
a)
a) < p1 < p2 + t(b
p1 > p2 + t(b
a)
a)
A Nash equilibrium including the two competing …rms exists only when total
revenue of the innovative …rm on the interval p1 2 [p2 t(b a); p2 + t(b a)]
is higher than its maximal pro…t on the interval p1 2 [c1 ; p2 t(b a)]. Unless,
we can not discuss any licensing regime since we will not …nd necessarily two
…rms on the linear city. We distinguish between two innovation sizes:
When 0 < " < 58 tl
optimal pro…ts are :
8
>
>
>
>
>
<
max
A
>
>
>
>
>
:
r
A
2
t (a
5
+ b), we have a = b = 2l , and innovative …rm
= tl + 32 " l
1
18t
r
A
=
A
=0
max
A
(3tl + ")2 +
If
"
6t
(3tl
") If
p1 < p 2
p2
t(b
If
=
1
9t
"2
3
tl"
2
+ 92 t2 l2 < 0 (since
t(b
a)
a) < p1 < p2 + t(b
p1 > p2 + t(b
=
63 2 2
tl
4
a)
a)
< 0)
We conclude that when innovation is small, a Nash equilibrium do not exist
for a royalty licensing and …rm A has not interest to be on the price interval
p1 > p2 t(b a) since being on the interval p1 < p2 t(b a) is better
and it can makes higher pro…ts than the other interval. Since in the interval
p1 < p2 t(b a) …rm B leave the market so we have no license here.
However, when 58 tl 25 t(a + b) < " < 4tl t(a + b), we have a =
b = l, and innovative …rm optimal pro…ts are :
12
l
2
and
8
>
>
>
>
>
<
>
>
>
>
>
:
r
A
4tl
max
A
=
1
tl
3
1
18t
r
A
=
A
=0
max
A
+ 23 " l
7
tl
2
=
t(a + b))
+"
2
If
+
"
6t
5
tl
2
"
If
If
1
9t
"2
5
tl"
4
+
25 2 2
tl
8
p1 < p 2
p2
t(b
t(b
a)
a) < p1 < p2 + t(b
p1 > p2 + t(b
> 0 (since "0 =
5
tl
4
a)
a)
< 0 < "00 = 52 tl =
Nash equilibrium exists here under royalty licensing for an intermediate innovation.
References
[1] Alcácer, J. and Chung, W., (2007) Location Strategies and Knowledge
Spillovers, Management Science Vol. 53, No. 5, May 2007, pp. 760–776
[2] Alderighi, M. and Piga, C.,The Circular City with Heterogeneous Firms (July,
13, 2008). Available at SSRN: http://ssrn.com/abstract=1159381
[3] Alderighi, M. and Piga, C., (2009) , On cost restrictions in spatial competition
models with heterogeneous …rms , Department of Economics , Discussion Paper
Series.
[4] Arrow, K., (1962). Economic welfare and the allocation of resources for
inventions. In: Nelson, R. (Ed.), The Rate and Direction of Inventive Activity.
Princeton University Press, Princeton.
[5] Cohen, J.P. and C.J. Morrison Paul, 2004, Agglomeration Economies and
Industry Location Decisions: The Impacts of Spatial and Industrial Spillovers
[6] Hussler C., Lorentz A., Ronde P. (2007), "Agglomeration and endogenous
absorptive capacities: Hotelling revisited", Jena Economic Research Papers,
WP2007-12.
[7] Long, N.G., Soubeyran, A., 1998. R&D spillovers and location choice under
Cournot rivalry. Paci…c Economic Review 3, 105–119.
[8] Mai, C.-C. and Peng, S.-K., 1999. Cooperation vs. competition in a spatial
model. Regional Science and Urban Economics 29, 463–472.
[9] Matsumura and Matsushima (2008) On patent licensing in spatial competition
with endogenous location choice
[10] Osborne, M. and Pitchik, C. (1987), equilibrium in a hotelling’s model of spatial
competition, Econometrica 55 (1987), pp. 911–922
[11] Paci, R. Usai, S. (2000) Externalities, knowledge spillovers and the spatial
distribution of innovation, GeoJournal (vol. 4, 2000) special issue “Learning
13
and Regional Development: Theoretical Issues and Empirical Evidence”. Ed.
Ron Boschma
[12] Piga, C., Theotoky, J. (2005) Endogenous R&D spillovers and locational choice
, Regional Science and Urban Economics 35 (2005) 127–139
[13] Pinkse, Slade, C Brett (2002), Spatial price competition a semiparametric
approach, - Econometrica, 2002
[14] Poddar, S. and Sinha, U.B., (2004). On patent licensing in spatial competition.
Economic Record 80, 208–218.
[15] Hussler C., Lorentz A., Ronde P. (2007), "Agglomeration and endogenous
absorptive capacities: Hotelling revisited", Jena Economic Research Papers,
WP2007-12.
[16] Wang, X.H., 1998. Fee versus royalty licensing in a Cournot duopoly model.
Economics Letters 60, 55–62.
[17] Wang, X.H., 2002. Fee versus royalty licensing in a di¤erentiated Cournot
duopoly. Journal of Economics and Business 54, 253–266.
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