This note studies the transfer of a cost-reducing innovation from an independent patent-holder to an asymmetric Cournot duopoly that has different unit costs of production. It is found that royalty licensing can be superior to fixed-fee licensing for the independent patent-holder.Cournot duopoly

Bill Yang

X. Henry Wang

English

Research Papers in Economics

On Technology Transfer to an Asymmetric Cournot Duopoly 
X. Henry Wang Bill Yang
University of Missouri−Columbia Georgia Southern University
Abstract
This note studies the transfer of a cost−reducing innovation from an independent
patent−holder to an asymmetric Cournot duopoly that has different unit costs of production.
It is found that royalty licensing can be superior to fixed−fee licensing for the independent
patent−holder.
We thank Associate Editor Rod Garratt for his helpful suggestions.
Citation: Wang, X. Henry and Bill Yang, (2004) "On Technology Transfer to an Asymmetric Cournot Duopoly." Economics
Bulletin, Vol. 4, No. 14 pp. 1−6
Submitted: September 7, 2004.  Accepted: October 26, 2004.
URL: http://www.economicsbulletin.com/2004/volume4/EB−04D40009A.pdf  1 
1     Introduction 
  Starting from the pioneering work of Arrow (1962), a large body of literature has developed 
studying the issue of technology transfer by means of licensing. Kamien (1992) gives an excellent review 
of the earlier literature. One of the most important results is that of Kamien and Tauman (1986). These 
authors found that an independent patent-holder always prefers fixed-fee licensing to royalty licensing. 
Empirically, it has been found that both royalty and fixed-fee licensing are prevalent.
1 By going beyond 
the basic model of Kamien and Tauman, several papers have recently found that royalty can be superior 
to fixed-fee for the patent-holder. Wang (1998) and Kamien and Tauman (2002) found this result by 
considering the patent-holder as an insider in the oligopolistic industry. Saracho (2002) found a similar 
result when the oligopolistic industry is manager-managed.
2    
  This note  builds  on  the original basic model of Kamien and Tauman in that there is an 
independent  patent-holder with a cost-reducing innovation licensing her innovation to a Cournot 
oligopolistic industry. For simplicity, we focus on a Cournot duopoly with a linear demand. Our 
departure from the basic model is that the duopoly firms do not have equal unit costs, different from that 
assumed in the basic model by Kamien and Tauman. Namely, we study licensing to a duopoly with cost 
asymmetry. The main finding of this note is that when the pre-licensing cost difference between the 
duopolists is large enough and the patent-holder’s innovation is small enough, royalty licensing can be 
superior to fixed-fee licensing for the patent-holder. Obviously, from an empirical point of view, unequal 
unit costs must be the rule rather than an exception. For the purpose of most theoretical studies, the 
equal cost assumption is simplifying and inconsequential to the model. The present note indicates that 
this assumption plays an important role in the study of licensing choices and relaxing it produces a result 
that vindicates the prevalence of both fixed-fee and royalty licensing in the real world. 
 
2     Basic Model 
  Consider a Cournot duopoly producing a homogeneous product with the (inverse) market 
demand p = a  - Q, where p denotes price and Q represents industry output. With their old 
technologies, firm 1 produces at constant unit production cost  1 c  and firm 2 produces at constant unit 
production cost  2 c .  It is assumed that  a c c < £ < 2 1 0 , i.e., firm 1 is the more efficient firm. The cost-
reducing innovation by an independent innovator creates a new technology that lowers any adopting 
firm’s unit cost by the amount of e. Obviously,  1 c e£ .  
  With constant unit costs  1 c  and  2 c , the Cournot-Nash equilibrium is straightforward to obtain. 
In equilibrium, the firms’ quantities are 
  ) , ( 2 1 1 c c q  = 
3
2 2 1 c c a + - ,   ) , ( 2 1 2 c c q  = 
3
2 1 2 c c a + - ,          (1) 
their profits are 
                                                                 
1 According to Rostoker (1984), royalty alone was used thirty-nine percent of the time, fixed fee alone thirteen 
percent, and royalty plus fixed fee forty-six percent, among the firms surveyed. Similar evidence has been reported by 
Macho-Stadler et al (1996), Jensen and Thursby (2001) and others. 
2 Wang (1998) is an extension of Marjit (1990) who studied technology transfer within a Cournot duopoly. Kamien 
and Tauman (2002) extended Wang (1998) to a Cournot oligopoly.    2 
  ) , ( 2 1 1 c c p  = 
9
) 2 (
2
2 1 c c a + - ,   ) , ( 2 1 2 c c p  = 
9
) 2 (
2
1 2 c c a + - ,        (2) 
and total industry output is 
  ) c , c ( Q 2 1  = 
3
c c a 2 2 1 - - .                (3) 
  For simplicity and without loss of generality in terms of making the main point of this note, we 
assume that both firms produce positive output levels in all possible cost configurations. This assumption 
entails that the less efficient firm 2 produces a positive output even when the more efficient firm 1 is the 
only adopter of the cost-reducing innovation. Namely,  ) , ( 2 1 2 c c q e -  > 0 or equivalently 
3 
  1 2 2 c c a + - < e .                  (4) 
  A licensing game consists of three stages. In the first stage, the independent patent-holder sets a 
fixed licensing fee or a  per unit output  royalty rate. In the second stage, the  duopolists  decide 
simultaneously whether to accept the offer from the patent-holder. In the last stage, the two firms engage 
in a noncooperative competition in quantities. The patent-holder sets a fixed licensing fee or royalty rate 
to maximize her total licensing revenue. 
 
3     Fixed-fee Licensing  
  We consider first licensing by means of a fixed fee only. Under fixed-fee licensing, the patent-
holder licenses her cost-reducing technology to either firm at a fixed fee F which is invariant of the 
quantity the licensee will produce using the new technology. The maximum license fee a firm is willing to 
accept is the one that makes it indifferent between licensing and not licensing the new technology. If firm 
2 does not license, the maximum fee firm 1 is willing to accept is equal to 
  ) , ( ) , ( 2 1 1 2 1 1 c c c c p e p - -  =  e e) 2 (
9
4
2 1 + + - c c a .          (5) 
If firm 2 licenses, the maximum fee firm 1 is willing to accept is equal to 
  ) , ( ) , ( 2 1 1 2 1 1 e p e e p - - - - c c c c  =  e ) 2 (
9
4
2 1 c c a + - .          (6) 
For firm 2, it is willing to pay 
  ) , ( ) , ( 2 1 2 2 1 2 c c c c p e p - -  =  e e) 2 (
9
4
1 2 + + - c c a .          (7) 
if firm 1 is not a licensee and 
  ) , ( ) , ( 2 1 2 2 1 2 c c c c e p e e p - - - -  =  e ) 2 (
9
4
1 2 c c a + - .          (8) 
if firm 1 is a licensee. Obviously, both firms are willing to pay more if the other firm is not a licensee, as 
indicated by the fact is (5) is larger than (6) and (7) is larger than (8). Since (5) is larger than (7), if the 
patent-holder sells only one license she will charge a fee equal to (5), the amount firm 1 is willing to pay. 
Since (8) is smaller than (6), the patent-holder has to set a fee equal to (8) in order to sell two licenses.
4 
  Based on the above results, we have the following proposition. 
 
                                                                 
3 When c 1 = c 2 = c, (4) becomes  ac e<-. Hence, this assumption essentially means that we consider non-drastic 
innovations. 
4 For simplicity, we assume that a firm will buy a license when it is indifferent between buying and not buying.   3 
Proposition 1. Under fixed-fee licensing, the patent-holder will license her innovation to firm 1 only for 
12 a4c5c e>+- and she will license to both firms for  12 a4c5c e<+-. 
 
Proof: The patent-holder’s licensing revenue is equal to (5) if she sells only one license (to firm 1) by 
charging a fixed fee equal to (5). Denote her revenue from selling one license by 
  L M  =  e e) 2 (
9
4
2 1 + + - c c a .                (9) 
The patent-holder’s licensing revenue is equal to two times (8) if she sells two licenses by charging a 
fixed fee equal to (8). Denote her revenue from selling two licenses by  
  LL M  =  e + - ) c c 2 a (
9
8
1 2 .                (10) 
The conclusion of the proposition is immediate by comparing  L M  and  LL M .  
 
  This proposition indicates that, under fixed-fee licensing, large innovations will be licensed to the 
more efficient firm (firm 1) and small innovations will be licensed to both firms.
5 
 
4     Royalty Licensing 
  Consider now licensing by means of a royalty only. Under a royalty licensing method, the 
patent-holder sets a fixed royalty rate r and the amount of royalty each firm pays will depend on the 
output level it will produce using the new technology. By adopting the new technology, firm 1’s unit 
production cost becomes  r c1 + e -  and firm 2’s unit production cost becomes  r c 2 + e - . Obviously, if 
r £e both firms will choose to license and if r >e neither firm will license. 
  Utilizing (3), for r £e the patent-holder’s licensing revenue is 
    ) r c , r c ( Q r 2 1 + e - + e -  = 
3
) r 2 2 c c a 2 ( r 2 1 - e + - - .          (11) 
Maximizing (11) with respect to r subject to the constraint that r £e implies 
 
12 r 2acc2
4
e ￿
￿
= --+e ￿
￿ ￿
      
12
12
2acc
if;
2
2acc
if.
2
--
e£
--
e>
 
  It is straightforward to verify that the assumption (4) implies that  2 / ) c c a 2 ( 2 1 - - £ e . Hence, 
the patent-holder’s optimal royalty rate is  r =e. Substituting this into (11) gives the patent-holder’s 
licensing revenue under royalty licensing, given as 
  R = 
3
) c c a 2 ( 2 1 e - - .                  (12) 
 
5     Fee vs. Royalty 
   We evaluate next the superiority of fixed fee licensing versus royalty licensing. It is summarized 
                                                                 
5 In particular, if c 1 = c 2 = c then the known result that all non-drastic innovations are licensed to both firms and 
drastic innovations are licensed to only one firm is implied.    4 
in the following proposition.
6 
 
Proposition 2. For the patent-holder, royalty licensing is superior to fixed-fee licensing if and only if 
4 / ) c 7 c 5 a 2 ( 2 1 - + < e  and  2 c  >  13 / ) c 11 a 2 ( 1 + . 
 
Proof. If  4 / ) c 7 c 5 a 2 ( 2 1 - + < e  then from (9) and (12), R >  L M . If  2 c  >  13 / ) c 11 a 2 ( 1 +  then from 
(10) and (12),  R >  LL M . Hence, if  4 / ) c 7 c 5 a 2 ( 2 1 - + < e  and  2 c  >  13 / ) c 11 a 2 ( 1 +  then  royalty 
licensing is superior to licensing via a fixed fee either to firm 1 only or to both firms. This proves the “if” 
part of the proposition.  
For the “only if” part, consider first the case:  12 a4c5c e>+-. By Proposition 1, in this case 
the patent-holder will license only to firm 1 under fixed-fee licensing. Thus, if royalty is superior to fixed-
fee we must have R >  L M  >  LL M . From (9) and (12), R >  L M  implies  4 / ) c 7 c 5 a 2 ( 2 1 - + < e . From 
(10) and (12), R >  LL M  implies  2 c  >  13 / ) c 11 a 2 ( 1 + . Consider next the case:  12 a4c5c e<+-. By 
Proposition 1, in this case the patent-holder will license to both firms under fixed-fee licensing. Thus, if 
royalty is superior to fixed-fee licensing we must have R >   LL M  >  L M . Again R >  L M  and R >  LL M  
imply, respectively,  4 / ) c 7 c 5 a 2 ( 2 1 - + < e  and  2 c  >  13 / ) c 11 a 2 ( 1 + . This completes the proof for the 
necessary part of the proposition. 
 
  The condition  4 / ) c 7 c 5 a 2 ( 2 1 - + < e  implies that for a small enough innovation royalty can be 
superior  for the  patent-holder. The condition  2 c  >  13 / ) c 11 a 2 ( 1 +  can be rewritten as  1 2 c c -  > 
11 / ) c a ( 2 2 - , indicating that for royalty to be better for the patent-holder the cost difference between 
the efficient firm and the inefficient firm has to be large enough relative to the difference between the 
maximum price consumers are willing to pay and the inefficient firm’s unit cost. Obviously, the last 
condition is impossible to hold when the firms are equally efficient, implying the known result that in a 
symmetric market fixed-fee licensing is preferred to royalty licensing by the patent-holder.  
  The intuition for the result in Proposition 2 is fairly straightforward. Under royalty licensing with 
symmetric costs ( 2 1 c c = ) as in the basic model by Kamien and Tauman (1986) or asymmetric costs 
( 2 1 c c < ) as in the present model, firms do not gain in efficiency post-licensing and the total industry 
output is unchanged post-licensing. As a result, the  patent-holder’s total licensing revenue is 
proportional to the size of her i nnovation as shown by (12). Under fixed-fee licensing, the licensing 
firm(s) become more efficient, as a result the patent-holder can exploit this gain in their efficiency by 
reaping a licensing revenue via fixed fee. This total revenue is always larger than that obtainable under 
royalty licensing when the firms are equally efficient, as shown in Kamien and Tauman (1986). 
However, the patent-holder’s power in appropriating this efficiency gain is mitigated when the firms are 
not equally efficient. This mitigation takes one of two forms. When licensing to both firms, the fee has to 
be at a level the less efficient firm is willing to accept. When the pre-licensing cost difference between 
                                                                 
6 Based on (3), total industry output is always higher under fixed-fee licensing than under royalty licensing since at 
least one firm produces at a lower unit cost post-licensing under fixed-fee licensing while both firms’ unit costs stay 
at the same level post-licensing under royalty licensing. It is implied immediately that consumers always prefer fixed-
fee to royalty licensing, as in the basic model by Kamien and Tauman (1986). 
   5 
the two firms is sufficiently large, this renders fixed-fee licensing less preferable to royalty licensing as 
demonstrated above by comparing (10) and (12). When licensing to only one firm, the potential benefit 
from licensing to the other firm is totally lost. This loss is more significant (relative to total licensing 
revenue) the smaller is the innovation. When the innovation is sufficiently small, licensing to one firm via a 
fixed fee becomes less preferable to licensing by royalty as demonstrated above by comparing (9) and 
(12). 
  Figure 1 depicts a graphic illustration for the comparison between fixed-fee licensing and royalty 
licensing. In this graph, a = 1 and the value of  e is fixed. The unit square box contains all possible cost 
configurations. The assumption  2 1 c c £  removes all points below the forty-five degree line and the 
assumption  0 c c 2 a 1 2 > + -  removes all points above the AB line, where point A corresponds to 
( 1 c , 2 c ) = (0, 0.5) and point B corresponds to ( 1 c , 2 c ) = (1, 1). Hence, the triangle OAB represents all 
cost configurations of concern.  At points C, D and E, respectively, ( 1 c , 2 c ) = (0, 2/13), (0, 
(24)/7 -e ) and (113/3 -e, 111/3 -e ). Based on Proposition 2, for cost configurations below the DE 
line royalty is better than licensing to one firm via a fixed fee, and for cost configurations above the CB 
line royalty is better than licensing to both firms via a fixed fee. Hence, the triangle CDE corresponds to 
all cost configurations such that royalty licensing is preferred to fixed-fee licensing by the patent-holder. 
Obviously, this triangle is non-vacuous if point D is higher than point C (i.e., e < 3/13). 
 
6     Concluding Remarks 
  This note has studied and compared licensing by means of a fixed fee and licensing by means of 
a royalty to a Cournot duopoly by an independent patent-holder with a cost-reducing innovation. It is 
found that licensing by means of a royalty may be superior to licensing by means of a fixed fee from the 
view point of the patent-holder when the duopolists have different unit costs of production.  
  We have maintained the assumption that the cost-reducing innovation lowers any adopting 
firm’s marginal cost by a fixed amount. An alternative assumption, that is perhaps at least as natural as 
the one we adopted, is the case where the innovation lowers any adopting firm’s marginal cost by a 
fixed proportion. The mechanics of the model under this alternative assumption is mostly the same as the 
one presented in the preceding sections with the following three noticeable facts. First, the main 
conclusion in this note that both royalty licensing and fixed-fee licensing can be optimal for the 
independent patent-holder when firms have different levels of marginal cost continues to hold. Second, 
there are three possibilities under fixed-fee licensing: licensing to the more efficient firm 1 only, the less 
efficient firm 2 only, and to both firms. Third, under royalty licensing, either the less efficient firm is 
licensed or both firms are licensed. The main reason that accounts for the additional possibilities under 
either fixed-fee or royalty licensing is that with proportional reduction in marginal costs, the less efficient 
firm always gets a larger reduction in marginal cost from the innovation than does the more efficient firm. 
As a result, the less efficient firm is willing to accept a higher per unit output royalty and in the case of 
fixed-fee licensing, it may reap a higher gain in profit than does the more efficient firm. 
 
 
   6 
 
 
 
1 c
2 c
2 1 c c =
 
   7 
References 
 
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inventions,” American Economic Review, 91, 240-259.  
Kamien, Morton, 1992. “Patent licensing,” Chapter 11 in Handbook of Game Theory, eds. R. J. 
Aumann and S. Hart.  
Kamien, Morton and Yair Tauman, 1986. “Fees versus royalties and the private value of a patent,” 
Quarterly Journal of Economics, 101, 471-491. 
Kamien, Morton and Yair Tauman, 2002. “Optimal licensing: the inside story,”  The Manchester 
School, 70, 7-15. 
Macho-Stadler, I., X. Martinez-Giralt and J. D. Pérez-Castrillo, 1996. “ The role of information in 
licensing contract design,” Research Policy, 25, 43-57. 
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Saracho, A.  I., 2002. “Patent licensing under strategic delegation,”  Journal of Economics & 
Management Strategy, 11, 225-251.  
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60, 55-62. 
 

On Technology Transfer to an Asymmetric Cournot Duopoly

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