In their chapter in Dynamic Competition and Public Policy (2001, Cambridge University Press), Burtis and Kobayashi never defined their model's discount rate, making replicating their simulation results difficult. Through our own simulations, we were able to verify their results when using a discount rate of 0.10. We also identified two new types of equilibria that the authors overlooked, doubling the number of distinct equilibria in the model.

Matthew John Holian

Neil N. H. Nguyen

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 4 
  
Intellectual property and antitrust limitations on contract: comment 
  
 
 
Matthew John Holian  
San Jose State University 
Neil N. H. Nguyen  
San Jose State University
Abstract 
In their chapter in Dynamic Competition and Public Policy (2001, Cambridge University Press), Burtis and Kobayashi 
never defined their model's discount rate, making replicating their simulation results difficult. Through our own 
simulations, we were able to verify their results when using a discount rate of 0.10. We also identified two new types 
of equilibria that the authors overlooked, doubling the number of distinct equilibria in the model.
We thank Alliance Business School for providing research support during Holian's visit to Bangalore, India, in July, 2010, Lydia Ortega for 
useful comments on our earlier draft, and an anonymous referee for excellent suggestions. The usual disclaimer applies. 
Citation: Matthew John Holian and Neil N. H. Nguyen, (2010) ''Intellectual property and antitrust limitations on contract: comment'', 
Economics Bulletin, Vol. 30 no.4 pp. 2680-2684. 
Submitted: Jul 18 2010.   Published: October 12, 2010. 
 
     1  Introduction 
 
Michelle Burtis and Bruce Kobayashi (2001) present an interesting model of the 
software industry.  In our view, what is most interesting about this model is that it 
incorporates aspects of both Cournot and Schumpeterian competition, as well as variables 
that correspond with the strength of copyright and contractual protection for intellectual 
property.  As the discussion surrounding the model demonstrates, the model is rich and 
relates to many issues in both antitrust and licensing.
2 
However, we have discovered some problems with their presentation.  Most 
seriously, the discount rate was never defined, making replicating their simulation results 
difficult.   Therefore we undertook our own simulations in an attempt to verify their 
results.  The good news is that we were able to verify their results when using a discount 
rate of 0.10.    But we also discovered two new types of equilibria that the authors 
overlooked, taking the number of distinct equilibria in the model from two to four. Two 
other minor issues, discussed below, also merit clarification.  
We begin by briefly describing the model, and then go through our corrections 
point by point.  We present our own simulation results, containing our newly identified 
equilibria, in the Appendix.  In the model there are two types of firms, originators and 
imitators.  Originators spend resources creating software programs and imitators spend 
resources copying these programs.  Let  O N  be the number of firms producing original 
software and  I N  be the number of firms producing imitation versions of the source code.   
Market demand is given by  I I O O q N q N Q P − − =1 ) (,  w h e r e   O q  and  i q  are the 
quantities produced by originators and imitators, and Q is the total quantity of software 
units produced and sold by both types of firms.  Assuming marginal cost for originator 
and imitators is  O a and  I a  respectively, it is straightforward to take the first-order 
conditions for profit maximization, solve for reaction functions assuming symmetry, and 
then to solve the resulting system of equations for the Cournot quantities for each type of 
firm. 
At the time the decision to invest in producing an original or imitation version of 
the source code, the profit functions for each type of firm  j  (where  I O j , = ) equal: 
 
j j j j F q a Q P − − = Π ] ) ( [  
where  O F  and  I F  are the fixed costs of original development and imitation, respectively. 
In order to generate the simulation results reported in Table 1, Burtis and 
Kobayashi (2001) made specific assumptions about howz , the level of copyright 
protection, andk , the level of contractual protection, affect the cost of producing original 
and imitation versions of the source code. Specifically, they assumed that 
 
 
z z FO 05 . 0 4 . 0 ) ( + =  
                                                 
2 Our discussion applies equally to the 2001 published version, and to the working paper version available 
at http://ssrn.com/abstract=210088,  which seem to be essentially identical.     z z FI 10 . 0 1 . 0 ) ( + =  
k k aO 001 . 0 01 . 0 ) ( − =  
. 01 . 0 01 . 0 ) ( k k aI + =  
If we assume that the static per-period profits are collected in perpetuity, the 
present value of profits net of the costs of authorship or imitation turn out to be 
 
O O I I I I O O F r N N a N N a − + + + + − = Π / )] 1 /( ) ) 1 ( 1 [(
2 *
 
I O I O O O I I F r N N a N N a − + + + + − = Π / )] 1 /( ) ) 1 ( 1 [(
2 *  
where r is the discount rate. 
The number of firms in the model is endogenously determined. Given the at least 
one version of the original source code exists, the marginal firm can choose to enter 
either with its own original version of the source code or with an imitation version of the 
source code. Thus, a distribution of firms is an equilibrium if the following set of 
conditions are met.  First, imitation versions cannot exist in the absence of at least one 
original  version of the source code.  Second, further entry by either an original or 
imitation firm cannot be profitable.  Third, no existing firm of one type can make higher 
profits by changing its decision on whether to produce an original or imitation version.  
Finally, all existing firms must expect that the present value of their investment in 
producing an original or imitation version of the software program is positive. 
As mentioned above, we have identified several problems with Burtis and 
Kobayashi’s (2001) presentation.  We will now go through our (five) corrections point by 
point.  First, there were errors in their equations (5) and (6).  These equations did not 
include the marginal costs  O a  and  I a .
3  Second, there were errors in their equation (14).  
Total welfare must include each originator’s and imitator’s profit in addition to the sum 
of all generated consumer surpluses.
4  Third, there was a minor typo in their equation 
(25).
5  Fourth, the discount rate was never defined, making replicating their simulation 
results difficult. Through our own simulations, we were able to verify their results when 
using a discount rate equal to 0.10.  Finally, while we verified that all of their proposed 
equilibria in fact met the equilibrium conditions, we also identified two new equilibria.  
These new equilibria are indicated by an “ * ” in Table 1 in the Appendix.  The cells in 
which these new equilibria exist are also shaded in Table 1.  
                                                 
3 These are the first-order conditions; the correct equations are: 
(5)  2 / ] ) 1 ( 1 [ O I I O O O a q N q N q − − − − =   
(6)  . 2 / ] ) 1 ( 1 [ I I I O O I a q N q N q − − − − =     
4 The correct equation for total (or gross) welfare is: 
(14)  . * * 2 / *) * *))( ( * 1 ( * I I O O I I O O N N r q N q N Q P TW Π + Π + + − =  
5 The equation, with the correct notation, is as we reported above:  (25)    k k aI 01 . 0 01 . 0 ) ( + = . Of all of our corrections our fifth, that Burtis and Kobayashi (2001) underreported 
both the number and variety of equilibia, is in our view our most valuable contribution.  
On the whole, our findings indicate not only that their model is correct, but we also show 
that it is more interesting—the model supports a wider variety of equilibria than the 
original authors realized.  In particular, they identified only two types of equilibria, and 
only one of these had imitator firm entry.  This is strange for a model of the software 
industry where imitation is a rather prevalent phenomenon.  However, the two additional 
equilibria that we discovered both have multiple imitator firms.
6  In one case there are 
two imitators and two originators, and in the other case there are three imitators and one 
originator.   
 
2  References 
 
Burtis, M. and Kobayashi, B. (2001)  “Intellectual Property and Antitrust Limitations on 
Contract” in Dynamic Competition and Public Policy: Technology, Innovation, and 
Antitrust Issues by J. Ellig, editor,  Cambridge University Press: New York, 229-263. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                 
6 To illustrate why these are in fact equilibria, consider first the equilibrium at z=1, k=8, where NO, NI=1, 
3.  This is an equilibrium, because the originator is making profit equal to 0.14, and imitators are making 
profit equal to 0.087 (i.e. both are making positive profits), further entry by an originator drives originator 
profit down (to -0.041), while further entry by an imitator drives imitator profit down (to -0.001); if an 
imitator switches to become an originator, his profit would go down to 0.07, while if the originator 
switches to become an imitator, his profit would fall to 0.139. The proof of the second equilibrium we 
discovered, at z=1, k=9, where NO, NI=2, 2, follows in a similar manner. 3  Appendix 
 
Table I.  Updated simulation results
7 
  A. (z = 0) 
FI = .10 
FO = .40 
B. (z = 1) 
FI = .20 
FO = .45
C (z = 2) 
FI = .30 
FO = .50
D. (z = 3) 
FI = .40 
FO = .55
E. (z = 4) 
FI = .50 
FO = .60
I. (k = 0) 
aI = .010 
aO = .010 
No Equilibrium  No Equilibrium No Equilibrium No Equilibrium  No Equilibrium
II. (k = 1) 
aI = .020 
aO = .0095 
No Equilibrium  No Equilibrium No Equilibrium No Equilibrium  No Equilibrium
III. (k = 3) 
aI = .030 
aO = .0090 
No Equilibrium  No Equilibrium  No Equilibrium No  Equilibrium No  Equilibrium 
IV. (k = 4) 
aI = .040 
aO = .0085 
No Equilibrium  No Equilibrium No Equilibrium No Equilibrium  NO, NI=3,0 
P = .256, Q = .74 
GW = 4.61 
NW = 2.81
V. (k = 5) 
aI = .050 
aO = .0080 
No Equilibrium  No Equilibrium No Equilibrium NO, NI=2,1 
P = .267, Q = .73  
GW = 4.61 
NW = 2.96
NO, NI=3,0 
P = .256, Q= .74  
GW = 4.61 
NW = 2.81
VI. (k = 6) 
aI = .060 
aO = .0075 
No Equilibrium  No Equilibrium No  Equilibrium NO, NI=3,0 
P = .256, Q = .74  
GW = 4.62 
NW = 2.97
NO, NI=3,0 
P = .256, Q = .74  
GW = 4.62 
NW = 2.82
VII. (k = 7) 
aI = .070 
aO = .0070 
No Equilibrium  No Equilibrium NO, NI=3,0 
P = .255, Q = .74  
GW = 4.62 
NW = 3.12
NO, NI=3,0 
P = .255, Q = .74  
GW = 4.62 
NW = 2.97
NO, NI=3,0 
P = .255, Q = .74  
GW = 4.62 
NW = 2.82
VIII. (k = 8) 
aI = .080 
aO = .0065 
No Equilibrium  NO, NI=1,3* 
P = 0.249, Q = .75  
GW = 4.27 
NW = 2.87 
 
NO, NI=3,0 
P = .255, Q = .75  
GW = 4.63 
NW = 3.13 
NO, NI=3,0 
P = .255, Q = .75  
GW = 4.63 
NW = 2.98 
NO, NI=3,0 
P = .255, Q = .75  
GW = 4.63 
NW = 2.83 
IX. (k = 9) 
aI = .090 
aO = .0060 
No Equilibrium  NO, NI=3,0 
P = .255, Q = .75  
GW = 4.63 
NW = 3.28 
 
NO, NI=2,2* 
P = .238, Q = .76  
GW = 4.42 
NW = 3.12 
 
NO, NI=3,0 
P = .255, Q = .75  
GW = 4.63 
NW = 3.13 
NO, NI=3,0 
P = .255, Q = .75  
GW = 4.63 
NW = 2.98 
NO, NI=3,0 
P = .255, Q = .75  
GW = 4.63 
NW = 2.83 
 
                                                 
7 At the suggestion of an anonymous referee, we attempted to replicate the gross welfare (GW) and net 
welfare (NW) calculations. In this process, we discovered the errors in their equations (14) as mentioned in 
footnote 4 above.  In addition, we found that Burtis and Kobayashi’s report of GW and NW are off by a 
magnitude of 10, suggesting they forgot to include the discounted present value of consumer surplus. This 
can be quickly verified as  GW = NW + NOFO + NIFI. 

Intellectual property and antitrust limitations on contract: comment

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Intellectual property and antitrust limitations on contract: comment

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