2007・35
On patent licensing in spatial competition
with endogenous location choice
Toshihiro Matsumura
Noriaki MatsushimaOn patent licensing in spatial competition with endogenous
location choice
Toshihiro Matsumura
Institute of Social Science, the University of Tokyo
Noriaki Matsushima∗
Graduate School of Business Administration, Kobe University
July 23, 2007

Abstract
Using a standard linear city model with two firms, we consider how their licensing
activities following their R&D investments affect the locations of the firms and the effort
levels of the R&D investments. Although recent studies show that R&D investmentsthat may cause a large cost differential between firms yield agglomeration of the firms,
we show that licensing activities after R&D investments always lead to the maximum
differentiation between the firms and mitigate the price competition. We also show thatthose licensing activities induce the socially optimal effort levels of R&D activities.

JEL classification: L13, O32, R32
Key words: licensing, oligopoly, R&D, location∗

Correspondence author: Noriaki Matsushima, Graduate School of Business Administration, Kobe University, Rokkodai 2-1, Nada, Kobe, Hyogo 657-8501, Japan. Phone: +81-78-803-6981, Fax: +81-78-803-6977,
E-mail: nmatsush@kobe-u.ac.jp1

1

Introduction

Since the seminal work of Hotelling (1929), the model of spatial competition, which is oneof the most important models of oligopoly, has been seen by many subsequent researchers
as an attractive framework for analyzing product differentiation.1 The major advantage ofthis approach is that it allows an explicit analysis of product selection. Of particular interestis the equilibrium pattern of product locations and the degree of product differentiation.
The original finding of Hotelling (1929) is that firms produce similar products (minimum
differentiation). d’Aspremont et al. (1979) consider two-stage location–price games. Theyshow that products are maximally differentiated when transport costs are quadratic. The
firms in their study never choose minimal differentiation, to avoid cutthroat competition in
the subsequent stage.
Recently, several researchers have introduced quality uncertainty into the duopoly location model. This uncertainty stems from R&D activities. Gerlach et al. (2005) and Christou
and Vettas (2005) provide interesting results concerning the relation between technologicalfluctuation (the uncertainty of the outcomes of R&D activities) and product positioning.
They mainly consider the following three-stage game: first, each firm chooses its location ina linear city; second, each firm’s product quality is probabilistically determined; and third,
each firm sets its price. They show it is possible that spatial agglomeration (minimum differentiation) appears in equilibrium while the basic structures of their models differ fromeach other. In Gerlach et al. (2005), if the product quality of a firm is zero (the projectis unsuccessful), the firm remains inactive in the market. In Christou and Vettas (2005),
each firm has the ability to produce its product. Depending on the quality difference determined probabilistically and the locations of the firms, the quantities supplied by the firms
are determined.2
1

See, d’Aspremont et al. (1979), Neven (1985), and Anderson et al. (1992), among others.

2

Some studies have already shown that minimum differentiation may appear in equilibrium. Price collusion

after firms have made location choices is considered in Friedman and Thisse (1993). Cooperation between
firms is considered in the form of information exchange through communication by Mai and Peng (1999).
de Palma et al. (1985) introduce unobservable attributes in brand choice and consider situations in whichcentral agglomeration does not induce the standard Bertrand competition with homogeneous products. Theyshow that sufficient heterogeneity between firms induces central agglomeration. Bester (1998) introduces2We suppose that a firm undertakes R&D to develop an advanced technology to produce
its product. We often observe that such a firm transfers its technology to its rivals, subject toa fee. That is, after the outcome of R&D is determined, firms having advanced technologies
often earn additional profits from licensing fees. Neither Gerlach et al. (2005) nor Christou
and Vettas (2005) discuss these licensing activities, which are potentially important from theviewpoint of product positioning.
One purpose of our paper is to consider how licensing activities affect the locations offirms. That is, we introduce the following stage into the models of Gerlach et al. (2005) andChristou and Vettas (2005): a firm that has a cost advantage over its rival might license its
technology after their model’s second stage.
We consider how licensing activities following R&D affect the product choices of firms.
There exist several modes of licensing: a royalty per unit of output produced with the transferred technology, a fixed fee that is independent of the quantity supplied with transferredtechnology, or a hybrid such as a two-part tariff. In our paper, we consider a royalty perunit of output produced with the transferred technology.3 Contrary to the results in Gerlach
et al. (2005) and Christou and Vettas (2005), we show that the maximum differentiation
always appears in equilibrium when each firm is able to transfer its own advanced technology
to its rival via licensing.
Since the seminal work of Arrow (1962), there is a vast literature (see Kamien (1992)),
which focuses on the licensing arrangement optimal for the patentee in a wide variety ofsituations. There are papers that analyze various aspects of patent licensing (for instance,
insider or outsider patentees, incentives for innovation, and market structures (monopoly,
oligopoly, and competitive)). Although most of the papers use a standard framework of priceand quantity competition, endogenous product positioning is not considered.4
vertical quality characteristics, asymmetric information about this quality between sellers and consumers,
and a limited number of repeat purchases by consumers. Bester (1998) finds that, in equilibrium, central
agglomeration appears if the number of repeat purchases is more than one and not too large. Gerlach et al.
(2005) and Christou and Vettas (2005) also deal with models of endogenous production costs (quality) withspatial competition. See, for instance, Brekke and Straume (2004) and Matsushima (2004, 2006).
3

Two-part tariff licensing is chosen by firms in our model.

4

In the literature of licensing, the interactions between the patentee and licensee(s) are usually discussed3

To our knowledge, Poddar and Sinha (2004) are alone in employing the Hotelling modelwith cost asymmetry in the analysis of licensing.5 Although they consider the optimal
licensing strategy of an outsider patentee as well as an insider, the locations of the firms are
exogenously given. We think, therefore, that the setting discussed here provides additional
insight for the licensing literature. In addition, there is another important difference between
this paper and Poddar and Sinha (2004). In our paper, the firm that holds the patent chooses
its price to maximize total profits (profits from its own products plus revenue from licensefee). On the other hand, Poddar and Sinha (2004) assume that the firm chooses its price so
as to maximize its profit from its own products and neglects license fee revenue (although
the firm maximizes total profit when it chooses the level of the license fee).6 We show that
one of their results on nonneutrality does not hold in a more plausible setting.
We also consider whether the level of R&D activity under such a licensing scheme isexcessive or insufficient from the viewpoint of social welfare. We show that the incentive
for R&D investment by the firm is consistent with the additional social benefit of R&D
investment. That is, efficient R&D investments is achieved by firms without any government
intervention.
The remainder of this paper is organized as follows. Section 2 presents the basic model.
Section 3 shows the result under the exogenous location model. Section 4 shows the result
under the endogenous location model. Section 5 shows the main result of the paper. Sections
6 and 7 discuss the levels of R&D investments, and Section 8 concludes.
and whether the licensee(s) are insider(s) or outsider(s) is important (see, for instance, Hernández-Murilloa
and Llobet (2006), Kamien et al. (1992), Kamien and Tauman (1984, 2002), Katz and Shapiro (1986), Muto
(1993), Sen and Tauman (2007), Wang (1998, 2002), and Wang and Yang (1999), among others).
5

Some papers on licensing include the role of product differentiation. See, for instance, Caballero-Sanz

et al. (2002), Faulı́-Oller and Sandonı́s (2002), Mukherjee and Balasubramanian (2001), Muto (1993), Wang
and Yang (1999). Those papers do not use location models except for Caballero-Sanz et al. (2002) which
treat a location model with exogenous location choices.
6

Poddar and Sinha (2004) do not explicitly state this assumption, but their result does not hold without

this assumption. See also footnote 9 of our paper.

4

2

The model

Consider a linear city along the unit interval [0, 1], where Firm 1 is located at x1 and Firm 2
is located at 1 − x2 . Without loss of generality, we assume that x1 ≤ 1 − x2 . Consumers are
uniformly distributed along the interval. Each consumer buys exactly one unit of the good,which can be produced by Firms 1 and 2. The utility of the consumer located at x is given
by:
(

ux =

−T (|x1 − x|) − p1
if bought from Firm 1,−T (|1 − x2 − x|) − p2 if bought from Firm 2,(1)

where T (·) represents the transport cost incurred by the consumer. T (·) is an increasing
function. For a consumer living at x(p1 , p2 , x1 , x2 ), where−T (|x1 − x(p1 , p2 , x1 , x2 )|) − p1 = −T (|1 − x2 − x(p1 , p2 , x1 , x2 )|) − p2 ,(2)

total utility is the same whichever of the two firms is chosen. Thus, the demand facing Firm
1, D1 , and that facing Firm 2, D2 , are given by:
D1 (p1 , p2 , x1 , x2 ) = min{max(x(p1 , p2 , x1 , x2 ), 0), 1},D2 (p1 , p2 , x1 , x2 ) = 1 − D1 (p1 , p2 , x1 , x2 ).

(3)

We now assume that one of the firms (denoted as Firm 1) has a cost-reducing innovation.
Assume preinnovation marginal costs of both firms are c1 = c2 = c. Firm 1’s cost-reducing
innovation lowers its marginal cost by d > 0, so that postinnovation, c1 = c − d and c2 = c.
Firm 1 now licenses its new technology to Firm 2 at a royalty rate (r). The total royalty
Firm 2 pays will depend on the quantity supplied by Firm 2 using the new technology. In thiscase, Firm 1’s marginal cost of production is c − d and Firm 2’s marginal cost of production
becomes c − d + r. The royalty r ≤ d, otherwise Firm 2 will not accept the contract.
The game runs as follows. In the first stage, both firms i choose their location xi ∈ [0, 1]
simultaneously. In the second stage, r is determined (Firm 1 offers the royalty r for Firm
2. Firm 2 chooses whether or not to accept it).7 In the third stage, the firms i choose theirprices pi ∈ [ci , ∞) simultaneously.8
7

The order of the first and second stages is interchangeable . If Firm 1 chooses r and then the firms

choose their locations, all our Section 3 and 4 results hold true.
8

Choosing pi < ci is weakly dominated by choosing pi = ci , so we assume that the lower bound of theprice is its cost.

5

3

Result: Exogenous locations

To clarify the effect of royalty licensing on the quantities supplied by the firms, we first
discuss the case in which their locations are exogenously given by x1 = x2 = 0. That is,
we assume that the firms locate at the edges respectively. We now label x(p1 , p2 , x1 , x2 )x(p1 , p2 ). The profit functions of the firms are:π1 = (p1 − (c − d))x(p1 , p2 ) + r(1 − x(p1 , p2 ))= (p1 − (c − d) − r)x(p1 , p2 ) + r,
π2 = (p2 − (c − d) − r)(1 − x(p1 , p2 )).
The first-order conditions are:∂π1∂p1

= x(p1 , p2 ) + (p1 − (c − d) − r)∂x(p1 , p2 )∂p1

= x(p1 , p2 ) − (p1 − (c − d) − r) ×
∂π2
∂p2

1
T 0 (1 − x(p1 , p2 )) + T 0 (x(p1 , p2 ))= 1 − x(p1 , p2 ) − (p2 − (c − d) − r) ×,

1
T 0 (1 − x(p1 , p2 )) + T 0 (x(p1 , p2 )).

The first-order conditions lead to:µ ¶11, x(p1 , p2 ) = .
22

p1 = p2 = c − d + r + T 0
The profits of the firms are:π1 =T 0 (1/2)
+ r,
2

π2 =

T 0 (1/2).
2(4)

We find that the quantities supplied by the firms do not depend on the level of the royalty
fee, r. We also find that Firm 2’s profit does not depend on r. To explain, when the licensing
firm sets its price, it takes into account the per unit licensing fee, r. r(1 − x) is the profit
from selling its own technology to the rival firm. When the licensing firm increases its output
level (x), the rival’s supply (1 − x) decreases and then the profit from licensing r(1 − x) also
decreases. The licensing fee can be interpreted as the opportunity cost to the licensing firm.99Note that the result that the quantities supplied by the firms do not depend on the level of the licensing

fee r is different from that in Poddar and Sinha (2004). Using the Hotelling model with cost asymmetry, they6

Since Firm 1’s unit cost (the real cost plus the opportunity cost) is c − d + r and Firm 2’sunit cost is c − d + r, both firms face the same unit cost, which leads to equal market share.
An increase in r increases per unit cost of each firm and raises equilibrium prices. This is
why the profit of Firm 2 does not depend on r.
This property is similar to that in Sappington (2005). Using a simple duopoly model,
he considers the problem of access charge and the entrant firm’s make-or-buy decision. The
vertically integrated incumbent firm charges the input price w to the entrant firm. He showsthat the level of w does not affect the quantities supplied by the firms.10
Since r ≤ d, Firm 1 naturally sets r = d. Since an increase in r does not reduce the profit
of Firm 2 and increases that of Firm 1, Firm 1 chooses the maximal royalty fee and Firm
2 has no reason to reject it. Note that this result does not depend on the distribution of
bargaining power between Firms 1 and 2.

4

Result: Endogenous locations

We now model endogenous firm locations. Before we solve the problem, we show a closely
related result provided by Ziss (1993). Using the Hotelling model with cost asymmetry,√he shows that pure strategy equilibria do not exist if and only if d ≥ (6 − 3 3)t (i.e., thecost difference between the firms d is large enough). We now show that when the efficient
firm is able to license its efficient technology to the inefficient rival, in all cases, maximum
differentiation appears in equilibrium. Obviously, the location equilibrium is quite different
from that in Ziss (1993).
consider the optimal licensing strategy of both an outsider as well as an insider patentee. Although Poddar
and Sinha (2004) present several interesting insights about optimal licensing strategies, they do not consider
the effect of the licensing fee on pricing strategy. Therefore, in their paper, the quantity supplied by thelicensed firm is (3 − r)/6, where r is a royalty rate (Poddar and Sinha 2004, p. 215). In our paper, we assume
the patentee is an insider (Firm 1 holds the patent). If we consider an outsider patentee, we can show thatthe patentee’s profit is d, which is exactly the same as the additional profit of insider patentee. This implies
that neutrality (both insider and outsider patentees have the same incentive for R&D) holds in our spatialmodel.
10

Note that the main proposition of Sappington (2005) is not the property already mentioned, but thefollowing one: regardless of the established price (w) of the upstream input, the entrant prefers to make (resp.
buy) the upstream input itself (resp. from the incumbent) when it (resp. the incumbent) is the least-costsupplier of the input.

7

To derive the location equilibrium explicitly, we assume that the transport costs of consumers are quadratic in distance. The assumption is similar to that in Ziss (1993). That
is:
(

ux =

−t(x1 − x)2 − p1
if bought from Firm 1,
2−t(1 − x2 − x) − p2 if bought from Firm 2.

(5)

For a consumer living at:x=

1 − x1 − x2
p2 − p1+,22t(1 − x1 − x2 )(6)

total utility is the same no matter which of the two firms is chosen. Given the values of x1 ,x2 , r, the profits of the firms are:π1 = (p1 − (c − d))x + r(1 − x)µ

1 − x1 − x2
p2 − p1= (p1 − (c − d) − r)+2
2t(1 − x1 − x2 )
π2 = (p2 − (c − d) − r)(1 − x)µ= (p2 − (c − d) − r)

¶

+ r,

(7)

¶

1 + x1 + x2
p1 − p2+.
22t(1 − x1 − x2 )(8)

The first-order conditions lead to:t(1 − x1 − x2 )(3 + x1 − x2 )3t(1 − x1 − x2 )(3 − x1 + x2 )= (c + r − d) +.
3

p1 = (c + r − d) +p2

Substituting the prices into the profit functions in (7) and (8), we have:
π1 =
π2 =

t(1 − x1 − x2 )(3 + x1 − x2 )2+ r,
18t(1 − x1 − x2 )(3 − x1 + x2 )2.
18

For the same reason discussed in the previous section, Firm 1 sets the level of r at r = d inthe second stage. Differentiating πi with respect to xi , we have:
∂π1
∂x1
∂π2
∂x2

t(1 + 3x1 + x2 )(3 + x1 − x2 )< 0,
18t(1 + x1 + 3x2 )(3 − x1 + x2 )
= −
< 0.
18
= −

8

The optimal locations of the firms are x1 = x2 = 0. That is, the maximum differentiation
appears in equilibrium. The profits of the firms are:
π1 =

t
+ d,
2

t
π2 = .
2

When we take into account licensing activity, the well-known maximum differentiation
result appears even though the cost differential between the firms is significant. The result
is quite different from that in Ziss (1993).
To discuss the difference between this model and that in Ziss (1993), we consider themixed strategy equilibria in Ziss (1993). For the nonexistence problem, we can show that two√
firms independently choose two edges of the linear city with equal probability if d ≥ (6−3 3)t(this is a sufficient condition). Under the mixed strategy equilibrium, the following four
location patterns appear as the firms’ location choices: (1) Firm 1 locates at 0 and Firm 2
locates at 1 (with probability 1/4); (2) Firm 1 locates at 1 and Firm 2 locates at 0 (withprobability 1/4); (3) Firm 1 locates at 0 and Firm 2 locates at 0 (with probability 1/4); and,
(4) Firm 1 locates at 1 and Firm 2 locates at 1 (with probability 1/4).
Under the former two patterns (the maximum differentiation, with probability 1/2), theprofits of the firms are as follows:2 (3t + d)

π1 =




18t
d−t

d < 3t,2 (3t − d)π2 =d ≥ 3t,




18t
0

d < 3t,(9)d ≥ 3t.

Under the latter two patterns (the minimum differentiation at the edge, with probability1/2), the profits of the firms are as follows:
π1 = d, π2 = 0.

(10)√
When d ≥ (6 − 3 3)t, given that the firms employ the above-mentioned mixed strategies,
the expected profits of the firms are:π1 =(3t + d)2 d(3t − d)2
+ , π2 =
.
36t
2
36t

(11)

For any d < 3t, profits are smaller than those in which the efficient firm licenses its technology
to the inefficient rival.
9

5Main result: R&D investments and location choice

To show the main result of our paper, we now consider the following four-stage game: first,
each firm chooses its location in a linear city; second, each firm chooses whether or not
it engages in R&D investment that probabilistically affects its production cost; third, oneof the firms that has a cost advantage over its rival chooses whether or not it licenses its
technology to the rival, and chooses the royalty fee when it licenses; and fourth, each firm
sets its price.11
We now assume that Firm 1 has a cost advantage after its investment in R&D and thatthe difference between the two firms’ marginal costs is d. We have already shown that, given
locations and marginal costs, if the advantaged firm licenses its technology, the profits of thefirms are:
π1L ≡
π2L ≡

t(1 − x1 − x2 )(3 + x1 − x2 )2+ d,
18t(1 − x1 − x2 )(3 − x1 + x2 )2.
18

If the advantaged firm does not license its technology, the profits of the firms are:

π1N

π2N



d − t(1 − x2 − x1 )(1 − x1 + x2 ),




≡

≡

if d ≥ t(1 − x2 − x1 )(3 − x1 + x2 ),(d + t(1 − x1 − x2 )(3 + x1 − x2 ))2, if d < t(1 − x1 − x2 )(3 − x1 + x2 ),18t(1 − x1 − x2 )







0,



if d ≥ t(1 − x1 − x2 )(3 − x1 + x2 ),2(t(1 − x1 − x2 )(3 − x1 + x2 ) − d), if d < t(1 − x1 − x2 )(3 − x1 + x2 ).

18t(1 − x1 − x2 )In any case, the advantaged firm has the incentive to license its technology for its rival in
the third stage. In the first and the second stages, each firm anticipates the profit functions
π1L and π2L .
The results in the second stage only affect on the value of d in π1L . The locations are
independent from the results in the second stage that are related to the value of d. The
equilibrium locations are similar to those in Section 4. Therefore, maximum differentiation
11

Again, the order of the license fee decision and location choice stages is interchangeable. Our result holds

if the location choices are made after the determination of r.

10

appears in equilibrium even though each firm undertakes R&D. The result is quite different
from those of Gerlach et al. (2005) and Christou and Vettas (2005).

6

Further discussion on R&D investments

We now explicitly introduce R&D decisions into the basic model. The unit cost of the
product for each firm is ci (i = 1, 2), which is determined by its investment. Each firmchooses whether to invest to reduce its own production costs. By its investment, firm i has
to incur the fixed cost of the investment F , and it reduces its marginal production cost at
ci = c − h (h ∈ [0, c̄]). h depends on a density function f (h)(≥ 0) that has the following
property: F (h) ≡Rh

0 f (t)dt and F (c̄) = 1. We assume that f (h) and F (h) are continuous anddifferentiable functions. After the firms invest, Nature determines c1 and c2 independently
and simultaneously
We can summarize the situation with the following payoff matrix.
Firm 2
Firm 1I
N

I
E[π1 (I, I)], E[π2 (I, I)]
E[π1 (N, I)], E[π2 (N, I)]N
E[π1 (I, N )], E[π2 (I, N )]E[π1 (N, N )], E[π2 (N, N )]To check the incentive to invest, we have to evaluate:(1) ∆1 ≡ E[π1 (I, N )] − E[π1 (N, N )], (2) ∆2 ≡ E[π1 (I, I)] − E[π1 (N, I)].
∆1 is the expected additional profit from the investment given that the rival firm does notinvest. ∆2 is the expected additional profit given that the rival firm does invest. If ∆i ≥ F ,the firm has the incentive to invest given that the i − 1 firm invests.
Case (1): Profit

As discussed earlier, if a firm has a cost advantage, it has the incentive

to license its own advanced technology for its rival. The firm earns additional profit d fromthe license, where d is the difference between the marginal costs of the firms. On the other
hand, the profit of a firm using the licensed technology is constant, t/2. Under Case 1, theexpected additional profit of the investing firm (∆a1 ) is:∆a1 =Z c̄ µ
0

v+

¶

t
t
f (v)dv − =
2
2
11

Z c̄
0

vf (v)dv.

Case (1): Welfare

We now suppose that the social planner anticipates the licensing

regime. Regardless of the results of R&D activities, the quantity supplied by each firm is
1/2 and the prices set by the firms are the same. Therefore, R&D only affects the production
costs of the firms. Under the licensing regime, the most efficient firm’s technology is available
for all firms. For instance, when an innovating firm’s marginal cost becomes c − h, then theother firm’s marginal cost also becomes c − h and the social benefit of the R&D is h × 1 = h(1 is the total quantity supplied by the firms). If one firm invests, the gross benefit of the
R&D investment is:∆W1a ≡
Comparison (1)Z c̄
0vf (v)dv.

The difference between ∆a1 and ∆W1a is 0. The incentive for R&D

investments by the firms is consistent with the additional social benefit of that R&D. Sociallyefficient R&D investment is achieved by the firms without any governmental intervention.
As mentioned earlier, when an innovating firm’s marginal cost becomes c − h, then theother firm’s marginal cost also becomes c − h and the social benefit of the R&D is h × 1 = h.
By licensing, the innovating firm can take the whole social benefit of the R&D, h. Therefore,
the incentive for R&D investment by the firm is consistent with the additional social benefitof that investment. This logic can be also applied to the following case.
Case (2): Profit

The more efficient firm earns additional profit d from the license, whered is the difference between the marginal costs of the firms. Under Case 2, the expected
additional profit of the second investing firm (∆a2 ) is:
∆a2

=
=

Z c̄ Z c̄
0

0

0

0

Z c̄ Z v(v − min{v, t})f (t)dtf (v)dv(v − t)f (t)dtf (v)dv.

v − min{v, t} is the level of the ex post cost advantage by the investment, where t is the levelof the rival firm’s cost reduction and v is the level of its own cost reduction. If v > t, thelevel is positive and the firm gains additional profit v − t, otherwise it is zero.

12

Case (2): Welfare

If the most efficient innovating firm’s marginal cost becomes c − h,then the other firm’s marginal cost also becomes c − h and the social benefit of the R&D is
h × 1 = h (1 is the total quantities supplied by the firms). If both firms invest, the grossbenefit of the R&D investment is:
∆W2a

=
=

Z c̄ Z c̄
0

0

0

0

max{v, t}f (t)dtf (v)dv

Z c̄ Z v

vf (t)dtf (v)dv +Z c̄ Z c̄
0tf (t)dtf (v)dv.

v

max{v, t} is the level of the efficient firm’s cost reduction. If v > t, the decrease in the social
marginal cost is v, otherwise, it is t. To derive the additional benefit of the second R&D
investment, we change the expression ∆W1a as follows:∆W1a =Z c̄ µZ v
00vf (t)dt +

Z c̄
v

¶

vf (t)dt f (v)dv.

The additional social benefit of the second R&D investment is:∆W2a − ∆W1a
Comparison (2)=Z c̄ Z c̄0v(t − v)f (t)dtf (v)dv.

The difference between ∆a2 and ∆W2a − ∆W1a is:∆a2 − (∆W2a − ∆W1a )=Z c̄ Z v
0

−

=
=

0
Z c̄ Z c̄
0

Z c̄
0

vf (t)dtf (v)dv +0µZc̄ Z v00

Z c̄ Z c̄
0v

tf (t)dtf (v)dv +vf (t)dtf (v)dv −vf (v)dv −

Z c̄
0

0
Z c̄ Z c̄
0tf (t)dt ×

vf (t)dtf (v)dv

Z c̄ Z c̄

0

Z c̄
0

v

¶

tf (t)dtf (v)dv

tf (t)dtf (v)dv
f (v)dv= 0.
The incentive for R&D investment by the firms is consistent with the additional social
benefit of the R&D investment. Efficient R&D investment is achieved by the firms without
any governmental intervention.

7

R&D investments with and without licensing

We now compare R&D incentives in the following two cases: (a) each firm licenses its technology whenever it has a cost advantage; (b) neither firm licenses its advanced technology.
13

The basic setting is similar to that in the former section, except for the following simplification. By its investment, firm i reduces its marginal production cost at ci = c − c̄ withprobability p; ci = c − c̄/2 with probability q; ci = c with 1 − p − q. After the firms invest,
c1 and c2 independently and simultaneously. As mentioned in the former section, to checkthe incentive to invest, we now evaluate:(1) ∆1 ≡ E[π1 (I, N )] − E[π1 (N, N )], (2) ∆2 ≡ E[π1 (I, I)] − E[π1 (N, I)].
∆1 is the expected additional profit from the investment given that the rival firm does notinvest. ∆2 is the expected additional profit given that the rival firm invests. If ∆i ≥ F , thefirm has the incentive to invest, given that i − 1 firm invests.
Case (1)Given that no firm invests in R&D activity, the difference between the (gross)marginal gains in the cases (a) and (b) is (note that c < 3t):c̄(4(12t − c̄)p + (24t − c̄)q)
> 0.
72t

(12)

In any case, the R&D incentive under the licensing case is stronger than that under the noncooperative case.12 The reason is simple. When the innovating firm licenses its technology,
it can earn the whole additional social gain. On the other hand, when the innovating firm
does not license its technology, it can earn at most the product of its quantity supplied andthe level of the reduction in its marginal cost.
Case (2)Given that one of the firms invests in R&D activity, the difference between the(gross) marginal gains in cases (a) and (b) is:c̄(12[2(2p + q) − 3(2p2 + 2pq + q 2 )]t − (4p + q − 2(2p + q)2 )c̄)
.
72t

(13)

The sign of the fraction depends on the values of p, q, t, and h. Figure 1 depicts the regionsin which the sign is negative and positive when c̄ = t/10.13 When p + q is not large enough,12This result does not depend on the distribution function of the success probability of R&D investments

f (h).
13

The shape does not depend very much on the value of c̄/t.

14

the R&D incentive under the licensing case is stronger than that under the noncooperativecase.
We consider a case in which p + q is large enough, that is, the probability of success in
R&D is large enough. In the licensing case, given the rival firm invests, the probability thatthe other firm has a cost advantage over its rival is small, because the probabilities of successare high enough to allow both firms to succeed in their R&D investments. For instance, if
the marginal cost of an investing firm is c − d with probability 1, due to the investment,
the additional benefit of the second R&D investment is zero, that is, the additional R&D
investment is meaningless. In the nonlicensing case, given the rival firm invests, if the otherfirm does not invest, it has a cost disadvantage with higher probability. Contrary to thelicensing case, the cost disadvantage diminishes the profit of the inefficient firm. To avoid
being the inefficient firm, each firm has a strong incentive to invest. For instance, if themarginal cost of an investing firm is c − d with probability 1, due to the investment, theadditional benefit of the second R&D investment is t/2 − (3t − d)2 /18 = d(6t − d)/18 (seeSection 4).

8

Concluding remarks

We have investigated the relationship between licensing activities and equilibrium locations in
a product differentiation model. We also discuss the welfare implication of R&D investment.
We show that the level of royalty fee does not affect equilibrium locations nor equilibrium
profit of licenser and that firms choose the maximal license fee. We show that licensing
activities after R&D always lead to maximum differentiation between firms and mitigate pricecompetition. Furthermore, we show that those licensing activities induce socially optimal
R&D.
Our results have a testable implication for the literature of licensing. We have shownthat an efficient firm has an incentive to transfer its advanced technology to its rival and thatthe transfer mitigates competition between these firms. In the setting, we have implicitly
assumed that the technology is transferable. On the other hand, we think that the results
derived by Gerlach et al. (2005) and Christou and Vettas (2005) are suitable for industries
15

in which technological transfers are difficult or not available. That is, spatial agglomerationthat leads to tough competition appears if technological transfers are difficult.
Given our results, we think that the following is an important research question: are
the price–cost margins of firms positively related to technological transferability, which, inturn, is positively related to propensities to license? If technologies are transferable, pricecompetition is moderate and firms are able to earn greater than normal profits. On the other
hand, if technologies are not transferable, tough competition appears and firms earn normal
or subnormal profits. Future research is required to test this problem.
In this paper, we assume that the licenser can transfer its advanced technology without
any transaction costs. However, some technologies are difficult to transfer, and firms canchoose whether they invest in nontransferable or transferable technology. We think thatcompetitive structure affects this choice and this is an important problem in the context of
R&D investment. This remains a topic for future research.
[2007.7.23 826]

16

1

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1

Figure 1: R&D incentives
Horizontal: p, Vertical: q, The shaded area: ∆a1 − ∆b1 < 0, The white area: ∆a1 − ∆b1 > 0

17

References
Anderson, S. P., de Palma, A., and Thisse, J.-F., 1992. Discrete choice theory of productdifferentiation. Cambridge, MA: MIT Press.
Arrow, K. J., 1962. Economic welfare and the allocation of resources for invention. In:
Nelson, R.R. (ed.), The Rate and Direction of Inventive Activity: Economic and Social
Factors, Princeton University Press, Princeton, 609–625.
Bester, H., 1998. Quality uncertainty mitigates product differentiation. RAND Journal of
Economics 29, 828–844.
Brekke, K. R. and Straume, O. R., 2004. Bilateral monopolies and location choice. Regional
Science and Urban Economics 34, 275–288.
Caballero-Sanz, F., Moner-Colonques, R., and Sempere-Monerris, J.J., 2002. Optimal licensing in a spatial model. Annales d’économie et de statistique 66, 257–279.
Christou, C. and Vettas, N., 2005. Location choices under quality uncertainty. Mathematical Social Sciences 50, 268–278.
d’Aspremont, C., Gabszewicz, J.-J., and Thisse, J.-F., 1979. On Hotelling’s stability incompetition. Econometrica 47, 1145–1150.
de Palma, A., Ginsburgh, V., Papageorgiou, Y.Y., and Thisse, J.-F., 1985. The principle ofminimum differentiation holds under sufficient heterogeneity. Econometrica 53, 767–
781.
Faulı́-Oller, R. and Sandonı́s, J. 2002. Welfare reducing licensing, Games and Economic
Behavior 41, 192–205.
Friedman, J.W. and Thisse, J.-F., 1993. Partial collusion fosters minimum product differentiation. RAND Journal of Economics 24, 631–645.
Gerlach, H.A., Rønde, T., and Stahl, K., 2005. Project choice and risk in R&D. Journal ofIndustrial Economics 53, 53–81.
Hernández-Murilloa, R. and Llobet, G., 2006. Patent licensing revisited: Heterogeneousfirms and product differentiation. International Journal of Industrial Organization 24,
149–175.
Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41–57.
Kamien, M.I., 1992. Patent licensing. Handbook of game theory, vol. 1. Elsevier Science
Publishers: Amsterdam, Netherland.
Kamien, M.I., Oren, S.S., and Tauman, Y., 1992. Optimal licensing of cost-reducing innovation. Journal of Mathematical Economics 21, 483–508.
18

Kamien, M.I. and Tauman, Y., 1984. The private value of a patent: a game theoretic
analysis. Journal of Economics 4, 93–118.
Kamien, M.I. and Tauman, Y., 2002. Patent licensing: the inside story. Manchester School70, 7–15
Katz, M.L. and Shapiro, C., 1986. How to license intangible property. Quarterly Journalof Economics 101, 567–89.
Mai, C.-C. and Peng, S.-K., 1999. Cooperation vs. competition in a spatial model. Regional
Science and Urban Economics 29, 463–472.
Matsushima, N., 2004. Technology of upstream firms and equilibrium product differentiation. International Journal of Industrial Organization 22, 1091–1114.
Matsushima, N., 2006. Vertical mergers and product differentiation, GSBA Kobe University
Discussion Paper Series: 2006-9.
http://www.b.kobe-u.ac.jp/publications/dp/2006/2006 9.pdf
Mukherjee, A. and Balasubramanian, N. 2001. Technology transfer in a horizontally differentiated product market. Research in Economics 55, 257–274.
Muto, S., 1993. On licensing policies in Bertrand competition. Games and Economic
Behavior 5, 257–267.
Neven, D., 1985. Two stage (perfect) equilibrium in Hotelling’s model. Journal of Industrial
Economics 33, 317–325.
Poddar, S. and Sinha, U.B., 2004. On patent licensing in spatial competition. Economic
Record 80, 208–218.
Sappington, D. E. M., 2005. On the irrelevance of input prices for make or buy decisions.
American Economic Review 95, 1631–1638.
Sen, D. and Tauman, Y., 2007. General licensing schemes for a cost-reducing innovation.
Games and Economic Behavior 59, 163–186.
Wang, X.H., 1998. Fee versus royalty licensing in a Cournot duopoly model. Economics
Letters 60, 55–62.
Wang, X.H., 2002. Fee versus royalty licensing in a differentiated Cournot duopoly. Journalof Economics and Business 54, 253–266.
Wang, X.H. and Yang, B., 1999. On licensing under Bertrand competition. Australian
Economic Papers 38, 106–119.
Ziss, S., 1993. Entry deterrence, cost advantage and horizontal product differentiation.
Regional Science and Urban Economics 23, 523–543.

19