Licensing in a Hotelling model with quadratic transportation costs

Licensing in a Hotelling model with quadratic
transportation costs
Fehmi Bouguezzi 1
University of Carthage
Abstract
This paper studies optimal licensing regimes in a linear Hotelling model where …rms
are located at the end points of the city and where the transportation cost is not
linear but quadratic. We study for that a more general cost function and we try to
compare the …ndings with the results of the linear cost. We …nd the same optimal
licensing regimes. A per unit royalty is optimal when innovation is not drastic and
no licensing is better when innovation is drastic. We also …nd that no licensing is
always better than …xed fee licensing.
Key words: Hotelling model, technology transfer, patent licensing, quadratic
transportation cost
JEL codes : C21, L24, O31, O32
1
Introduction and model
Some authors discussed quadratic transportation costs in their models especially in the Hotelling and Salop models. Among those authors we …nd Egli
(2007) who studied equilibrium strategies in a Hotelling model where there
are two transportation cost functions, a linear and a quadratic. D’Aspremont
et Al (1979) studied optimal locations in a Hotelling model. They …nd that
with quadratic transportation costs, the two sellers will move to the two end
points of the city. Hinloopenyand et al (2013) studied optimal locations of two
1
Email : [email protected]
1
…rms when the transportation costs are linear then quadratic. Arguedas et
al (2007) studied the conditions of non existence of an equilibrium when the
transportation costs are quadratic. However, I could not …nd any study on
the e¤ect of the structure of the transportation costs on the optimal licensing
regime. So I try in this paper to whether the optimal licensing regimes found
in the Hotelling model with linear costs will stay true even in the quadratic
model or not.
I use the same model of the linear city à la Hotelling and I try to compare
the total revenues of the patent holding …rm when consumers pay a quadratic
transportation cost to move to one of the two …rms. I suppose that consumers
are distributed uniformly in the city and that …rms are located at the two end
points.
buy the product of firm 1
0
firm 1
buy the product of firm 2
x
l
firm 2
consumers
Fig 1 : The model
I suppose that the quadratic transportation cost function is the following :
C(d) = td + td2
Where t denotes the unit transportation cost, d the distance separating the
consumer and the …rm. The variables and will be used to compare the
results when the transportation costs is linear C(d) = td (for
= 1 and
= 0), then when the transportation costs is merely quadratic C(d) = td2
(for = 0 and = 1).
2
− U1
− U2
Total cost for a
consumer buying
from firm 2
Total cost for a
consumer buying
from firm 1
p2 + αt (l − x ) + βt (l − x ) 2
p1 + αtx + βtx 2
p2
p1
l
x
0
Fig 2 : Utility functions
The utility function contains in the following graphic a linear transportation
cost for = 1 and = 0. We …nd here the same model discussed in Poddar
and Sinha (2004)
− U2
− U1
Total cost for a
consumer buying
from firm 1
Total cost for a
consumer buying
from firm 2
p2 + t (l − x )
p1 + tx
p2
p1
0
x
linear transportation cost
α = 1, β = 0
Fig 3 : Linear transportation costs
3
l
The following utility functions contain here a merely quadratic transportation
cost for = 0 and = 1:
− U2
Total cost for a
consumer buying
from firm 1
− U1
Total cost for a
consumer buying
from firm 2
p2 + t (l − x ) 2
p1 + tx 2
p2
p1
l
x
0
quadratic transporta tion cost
α = 0, β = 1
Fig 4 : Merely quadratic transportation costs
The utility function of the consumer located in x ad buying the product of
the …rm 1 is : U1 = p1
tx
tx2
The utility function of the consumer located in x ad buying the product of
the …rm 2 is : U2 = p2
t(l x)
t(l x)2
The location of the marginal consumer is x~ and veri…es the equality between
the two utilities : x~ = 2t( 1+l ) (p2 p1 + tl + tl2 )
The demand function of …rm 1 is :
D1 =
D1 =
8
>
>
>
l
>
>
<
x~
>
>
>
>
>
: 0
8
>
>
>
l
>
>
<
2t(
>
>
>
>
>
: 0
tl2
if
p1 < p 2
tl
if
p2
tl2 < p1 < p2 + tl + tl2
if
p1 > p2 + tl + tl2
tl
if
1
+l )
(p2
p1 + tl + tl2 ) if
if
4
tl2
p1 < p 2
tl
p2
tl2 < p1 < p2 + tl + tl2
tl
p1 > p2 + tl + tl2
The demand function of …rm 2 is :
8
>
>
>
0
>
>
<
D2 = > l
>
>
>
>
: l
x~
8
>
>
>
0
>
>
<
p1 < p 2
tl
if
p2
tl2 < p1 < p2 + tl + tl2
if
p1 > p2 + tl + tl2
tl
if
1
2t( +l )
D2 = >
>
tl2
if
>
>
>
: l
(p1
p2 + tl + tl2 ) if
if
tl2
p1 < p 2
tl
p2
tl2 < p1 < p2 + tl + tl2
tl
p1 > p2 + tl + tl2
DA
DB
DA
DB
l
l
0
0
p2
c1
p2 − αtl − βtl 2
p1 < p2 − αtl − βtl 2
p2 + αtl + βtl 2
p1 − p2 < αtl + βtl 2
p1
p1 > p2 + αtl + βtl 2
Fig 5 : Demand functions
For each demand function, we have three di¤erent intervals depending on the
price. An interval where the two …rms are active on the market and two other
intervals where one of the …rms is active. In fact, the two …rms are active when
jp1 p2 j < tl + tl2 :
When jp1 p2 j > tl + tl2 , only one …rm will stay in the market and all the
consumers will buy its product.
The …rm 1 have the whole demand when the total costs for the farthest consumer (in l) who buys its product which includes the price and the transportation is lower than the product price of the form 2 ie p1 + tl + tl2 < p2 :
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As well, the …rm 2 will have the whole demand when its price and the transportation cost of the farthest consumer (located in 0) is lower than the price
of the product of …rm 1 ie p2 + tl + tl2 < p1 .
When the two …rms are active on the market ie when jp1
the pro…t functions are :
1
= (p1
c1 ) D1 = (p1
c1 ) 2t(
1
+l )
(p2
p1 + tl + tl2 )
2
= (p2
c2 ) D2 = (p2
c2 ) 2t(
1
+l )
(p1
p2 + tl + tl2 )
p2 j < tl + tl2 ,
Maximizing the two pro…ts with respect to the prices, we …nd :
8
>
<
>
:
@ 1
@p1
=0
@ 2
@p2
=0
=)
8
>
<p =
1
tl + tl2 + 31 (2c1 + c2 )
>
:p =
2
tl + tl2 + 31 (c1 + 2c2 )
To …nd the prices at the equilibrium, we looked for the price reaction functions
R1 (p2 ) and R2 (p1 ).
The …rst derivative
@ 1
@p1
The second derivative
= 0 gives p1 = R1 (p2 ) = 21 (p2 + c1 + tl + tl2 )
@ 2
@p2
= 0 gives p2 = R2 (p1 ) = 21 (p1 + c2 + tl + tl2 )
The intersection between the two reaction functions represents the couple
(p1 ; p2 ) of the prices at the equilibrium.
p2
R1 ( p2 )
R2 ( p1 )
1
*
p2 = αtl + β tl 2 + ( c1 + 2c2 )
3
1
( c2 + αtl + βtl 2 )
2
p1
1
*
p1 = αtl + βtl 2 + ( 2c1 + c2 )
3
1
( c1 + αtl + βtl 2 )
2
6
Fig 6 : Reaction functions
The demand function of the two …rms are :
D1 =
1
6t( +l )
(c2
c1 + 3 tl + 3 tl2 )
D2 =
1
6t( +l )
(c1
c2 + 3 tl + 3 tl2 )
Replacing in the pro…ts functions of the …rms 1 and 2, we …nd the pro…ts at
the equilibrium :
2
1
=
1
18t( +l )
(c2
c1 + 3 tl + 3 tl2 )
2
=
1
18t( +l )
(c1
c2 + 3 tl + 3 tl2 )
2
In the rest of the paper we will study the case where jp1 p2 j < tl + tl2 ie
1
(c1 c2 ) < tl + tl2 because we could not discuss licensing when only one
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…rm is on the market. We need to have two active …rms unless there could not
be any licensing.
When there’s no licensing (c1 = c " and c2 = c) we will get " < 3 tl + 3 tl
(non drastic innovation). For a drastic innovation, only the patent holding …rm
will stay on the market since it uses alone its non licensed new technology.
For a …xed fee or a per unit royalty licensing (c1 = c2 = c "), the condition
of two …rms is always veri…ed since tl + tl2 is always positive.
2
No licensing
In the no licensing regime, the marginal production cost of the patent holding
…rm is c1 = c ": The marginal cost of the …rm using the old technology is
c2 = c:
The price functions are the following :
p1 = t l 2 + t l + c
2
"
3
p2 = t l 2 + t l + c
1
"
3
the …rm 2 should sell its product with a price which is higher than its marginal
cost ie when " < 3 tl+3 tl2 which corresponds to a non drastic innovation. In
this case the two …rs are active on the market and demands are the following:
D1 =
1
6t( +l )
(3 tl + 3 tl2 + ")
7
D2 =
1
6t( +l )
(3 tl + 3 tl2
")
The corresponding pro…ts are :
2
1
=
1
18t( +l )
(3 tl + 3 tl2 + ")
2
=
1
18t( +l )
(3 tl + 3 tl2
2
")
When the innovation is drastic ie the size of the innovation is such that " >
3 tl + 3 tl2 , only one …rm will be active. The …rm using the old technology
will leave the market and the patent holding …rm becomes a monopoly. Its
price will be p1 = c ( tl + tl2 ). It will fox its price such that the farthest
consumer located in l will pay the marginal unit cost minus the transportation
cost from his location to the location of the patent holder.
The demand function of the two …rms are : D1 = l and D2 = 0 since …rm 1
will have the whole demand.
The pro…ts of the two …rms are :
3
NL
1
= ("
tl2 ) l and
tl
NL
2
=0
Fixed fee licensing
In the …xed fee licensing regime, the two …rms use the new technology and
their production unit costs are the same : c1 = c2 = c ".
The prices of the two …rms are : p1 = ( tl + tl2 + c
") and p2 = ( tl + tl2 + c
The demand of the two …rms are :
D1 =
1
6t( +l )
(3 tl + 3 tl2 ) and D2 =
F
1
The pro…ts are :
=
1
18t( +l )
1
6t( +l )
(3 tl + 3 tl2 )
2
(3 tl + 3 tl2 ) and
F
2
=
1
18t( +l )
2
(3 tl + 3 tl2 )
The …xed fee that should be paid by …rm 2 depends on the size of the innovation. In fact we have to distinguish between the cases where the innovation
is non drastic and the case where the innovation is drastic.
When the innovation is non drastic (" < 3 tl + 3 tl2 ), the …xed fee is :
F =
F
2
NL
2
=
1
18t( +l )
(6 tl + 6 tl2
") "
The total revenue of the patent holder will be :
F
1
=
F
1
+F =
1
18t( +l )
2
(3 tl + 3 tl2 ) +
8
1
18t( +l )
(6 tl + 6 tl2
") "
")
When the innovation is drastic (" > 3 tl + 3 tl2 ), the …xed fee is :
F =
NL
2
F
2
=
1
18t( +l )
2
(3 tl + 3 tl2 )
and the total revenue of the patent holder is :
F
1
=
F
1
+F =
1
9t( +l )
2
(3 tl + 3 tl2 )
Proposition 1 non licensing is always better than …xed fee licensing when
transportation costs are quadratic independently of the innovation size. We
…nd here the same result as in the Hotelling model with linear transportation
costs.
PROOF. for a non drastic innovation " < 3 tl + 3 tl2
F
1
NL
1
=
1
"2
9t( +l )
<0
for a drastic innovation " > 3 tl + 3 tl2
F
1
4
NL
1
= l (2 tl + 2 tl2
") < 0
A per unit royalty licensing
The technology transfer here is done through a per unit royalty. The production costs of the two …rms are : c1 = c " and c2 = c " + r where r denotes
the per unit royalty which must be positive and lower than the size of the
innovation ".
Demands of the two …rms are : D1 =
1
( r + 3 tl + 3 tl2 )
6t( +l )
The pro…ts are :
1
=
1
18t( +l )
1
6t( +l )
(r + 3 tl + 3 tl2 ) and D2 =
2
(3 tl + 3 tl2 + r) and
The total revenue of the patent holder is :
r 6t( 1+l ) ( r + 3 tl + 3 tl2 )
r
1
=
2
1 +rD2
=
=
1
18t( +l )
1
18t( +l )
(3 tl + 3 tl2
15
tl (
4
+ l) and r =
15
tl (
4
The total revenue of the …rm 1 will be :
9
+ l ) if " >
15
tl (
4
2
(r + 3 tl + 3 tl2 ) +
Maximizing this total revenue with respect to r we …nd two optimal royalties
depending on the size of the innovation :
r = " if " <
2
r)
+ l)
r
1
=
1
18t( +l )
r
1
=
33 2
tl
16
2
(" + 3 tl + 3 tl2 ) +" 6t(
( + l) if " >
15
tl (
4
1
+l )
( " + 3 tl + 3 tl2 ) if " <
15
tl (
4
+ l)
+ l)
Proposition 2 A per unit royalty licensing regime is optimal when innovation
is non drastic while no licensing is better for a drastic innovation. We …nd the
same optimal licensing regimes as in a linear transportation cost model where
the cost function equals td ( = 1 et = 0) and the same results also for a
merely quadratic transportation cost function where the function equals to td2
( = 0 et = 1).
PROOF. For a non drastic innovation " < 3 tl + 3 tl2
r
1
NL
1
=
1
"
6
1
t( + l)
+ 12 tl ( + l ) " > 0
For a drastic innovation " > 3 tl+3 tl2 we distinguish between two sub cases:
3 tl + 3 tl2 < " < 15
tl ( + l) and " > 15
tl ( + l) :
4
4
If 3 tl + 3 tl2 < " <
r
1
NL
1
If " >
r
1
5
=
1
18t( +l )
15
tl (
4
NL
1
=l
15
tl (
4
+ l)
(2" + 9 tl + 9 tl2 ) (3 tl + 3 tl2
") < 0
+ l)
49
16
tl +
49
16
tl2
" <0
Conclusion
We tried in this paper to discuss optimal licensing regimes in a Hotelling
model when the transportation cost is not linear but quadratic. We built for
that a linear Hotelling model where consumers are distributed uniformly and
where the two competing …rms are located at the two end points of the city.
Results show that we …nd the same optimal licensing regimes of the linear
transportation cost function. A per unit royalty is always optimal for a non
drastic innovation while no licensing is better when the innovation is drastic.
References
[1] Arguedas, C., Hamoudi, H., Saez, M., 2007, Equilibrium Nonexistence in Spatial
Competition with Quadratic Transportation Costs, Universidad Autónoma de
10
Madrid (Spain), Department of Economic Analysis (Economic Theory and
Economic History), Series Working Papers in Economic Theory number 2007/01.
[2] D’Aspremont et al., 1979, On Hotelling’s Stability in Competition, Econometrica,
47 (5), pp. 1145–1150
[3] Egli, A. 2007, Hotelling’s Beach with Linear and Quadratic Transportation Costs:
Existence of Pure Strategy Equilibria, Australian Economic Papers, vol. 46, issue
1, pages 39-51
[4] Hinloopenyand, J., Martinez, S., Costly location in Hotelling duopoly, 2013,
Tinbergen Institute Discussion Papers with number 13-101/VII
[5] Poddar, S., Sinha, U., 2004, On patent licensing in spatial competition, Economic
Record, Volume 80, Issue 249, pages 208–218
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