In innovative races with winner takes all, leading firms invest less than each follower, given exogenous entry (Reinganum, 1985). But with endogenous entry this result is reversed (Etro, 2004). It is argued here that sharing of rewards between the players may alter these predictions.Firms; Free entry; Innovation; IT; Market sharing; Patent race; Prediction; Predictions;

De Bondt, Raymond

Vandekerckhove, Jan

English

Research Papers in Economics

Innovation by leaders without winner-take-all
t
Raymond De Bondt and Jan Vandekerckhove
DEPARTMENT OF MANAGERIAL ECONOMICS, STRATEGY AND INNOVATION (MSI)
Faculty of Economics and Applied Economics
MSI 0709  1 
 
INNOVATION BY LEADERS 
WITHOUT WINNER-TAKE-ALL 
 
 
by Raymond De Bondt and Jan Vandekerckhove° 
 
Catholic University of Leuven 
Department of Managerial Economics, Strategy and 
Innovation  
Naamsestraat 69, 3000 Leuven 
Belgium 
 
June 2007 
 
Abstract: In innovative races with winner takes all, 
leading firms invest less than each follower, given 
exogenous entry (Reinganum, 1985). But with endogenous 
entry this result is reversed (Etro, 2004). It is argued 
here that sharing of rewards between the players may 
alter these predictions.  
 
Keywords: Patent Race, Free Entry, Innovation, Market 
Sharing 
 
JEL Classification Numbers: O31, L13 
 
1. Introduction 
 
A central feature of the neo-Schumpeterian analysis is 
the focus on the innovative role of first movers and 
followers. In so called race settings with technological 
uncertainty and winner-take-all, the leader invests less 
than each follower (Reinganum, 1985). This finding is 
consistent with empirical findings that, on average, 
challengers tend to invest more to enter a new market 
than incumbents (e.g. Czarnitzki and Kraft, 2004).    
 
Refinements of the hypotheses are possible. With 
endogenous entry a leading firm anticipates that the 
equilibrium number of entrants will be affected by its 
own efforts (Etro, 2004). The first mover engages in more 
efforts than each of the followers, given winner-take-
all. But what if the players have to share payoffs 
because of spillovers?   
 
 
2. The model 
 
Results on innovative races without winner-take-all have 
been analyzed by Stewart (1983) in a setting with   2 
simultaneous moves of all rivals. Here a stochastic 
leader-follower setting with market sharing is looked at.  
Let ￿ be a market sharing parameter (￿ ￿ 1). Two cases 
with asymmetric sharing define the minimum value of ￿ and 
the notation of payoffs, see Table 1.  
 
 
A 
 
Leaders may share 
with  
all entrants 
 
B 
 
Sharing may occur 
among all 
entrants 
 
 
 
 
 
 
One leader and 
n entrants 
(followers) 
Winner takes all: 
1 s =  
Equal sharing: 
1
1 n
s =
+  
 
Winner takes all: 
1 s =  
Equal sharing: 
1
n
s =  
P1
L 
(leaders 
wins) 
 
P s ´  
 
P  
 
 
Payoff 
leader  P2
L 
(entrant 
wins) 
 
0 
 
0 
P1
i
 
(entrant 
wins) 
 
P  
 
P s ´  
P2
i 
(leader 
wins) 
 
1
P
n
s - ￿ ￿ ´ ￿ ￿
￿ ￿
 
 
0 
 
 
 
 
 
Payoff 
entrant i 
 
P3
i 
(other 
entrant 
wins)  
 
 
0 
 
 
1
1
P
n
s - ￿ ￿ ´ ￿ ￿ - ￿ ￿
 
 
Table 1. Payoffs in two races with asymmetric market 
sharing. 
 
Firm L is a Stackelberg leader in the innovative race and 
firms 1,2,…,n are entrants in the industry and followers.  
The  development  process  is  stochastic  with  the 
probability  of  success  of  any  firm  j  by  time  t  being 
equal to  ( ) 1
j h x t
e
- ´
- , where h(xj) is the development intensity 
and xj is the development intensity selected by firm j, at 
a cost of xi euros per unit of time, with j=L,1,2,..,n. 
Moreover, h’(.)>0 and h”(.)<0. F is a fixed cost for each   3 
player, with  0 P F s ´ - > . The expected value of discounted 
profits for the leader with interest rate r is: 
 
 
( ) ( )
( ) ( )
p ￿
=
￿
=
´ + ´ + -
= -
+ +
1 2
1
1
n L L j L L
j L
n j L
j
h z P h z P z
V F
r h z h z
         (1)                                 
 
and for each of the entrants i=1,2, …,n: 
 
 
( ) ( ) ( )
( ) ( )
￿
=
¹
￿
=
´ + ´ + ´ -
= -
+ +
1 2 3
1
1
n i i L i j i i
j
j i i
n j L
j
h z P h z P h z P z
V F
r h z h z
       (2) 
 
The discussion hereafter focuses on parameter values with 
v
L>0 and V
i￿0. 
 
A symmetric equilibrium can be looked at with zi=z, for 
all i. Concavity and existence conditions are assumed to 
be satisfied. 
 
With exogenous entry the number of entrants is taken as 
given. The efforts of the followers are always strategic 
complements to the efforts of the leaders in setting B. 
In setting A, however, they are strategic complements for 
little sharing (high ￿) but may be substitutes for high 
spillovers and more equal sharing (low ￿).  
 
In a long run symmetric equilibrium V
i(z,z
L,n*)=0, with n* 
the endogenous number of entrants. Combining this from 
(2) with the first order condition for an entrant allows 
verifying that:  
 
Case A:  ( ) ( ) ' 1 h z P F ´ - =           (3) 
Case B:  ( ) ( ) ' 1 h z P F s ´ ´ - =            (4) 
 
So, in both cases, z is independent of the efforts of the 
leader z
L (Etro, 2004). The leader will however anticipate 
that n* may be influenced by its own effort.  
 
 
3. Comparing leader and entrant efforts.  
 
The comparison of leader and follower efforts is driven 
by the sign of the function g(x) with g(z
L)=0 and g’(.)<0 
in view of the concavity of V
L.  
 
Case A.   4 
( ) A g x =
( ) ( ) ( )
( )
' 1
L
A
h x P r n h x h z
h x P x
s
y s p
￿ ￿ ￿ ￿ ´ ´ - ´ + ´ + ￿ ￿ ￿ ￿ ￿ ￿
￿ ￿ - ´ ´ ´ + - ￿ ￿
      (5) 
with  ( )
( ) ( )
y
¶ ´
= +
¶
'
A
L
n h z
h x
z .  (6) 
 
Case B. 
( ) ( ) ( ) ( ) ( ) y p ￿ ￿ ￿ ￿ ￿ ￿ = ´ - ´ + ´ + - ´ ´ + - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿
' 1 L B
B g x h x V r n h x h z h x V x   (7) 
 
  with  ( )
( ) ( )
y
¶ ´
= +
¶
'
B
L
n h z
h x
z .          (8) 
 
 
Moreover, the following stability condition is assumed: 
( ) ( )
' 1 0 L
n z
h z
z
y · ￿ ￿ ¶ ´
= ´ + > ￿ ￿ ￿ ￿ ¶ ￿ ￿
.           (9) 
 
With endogenous entry,  0 L
z
z
¶
=
¶  and  ( ) ( )
*
' ' 0
n
h z h z
z
y · ¶
= + ´ >
¶ , so 
  ( )
( ) ( ) ( )
( ) ( )
*
*
' 1
'
1
A h z h x V n F
h x
h x V n F h z
n
s
y
s
￿ ￿ ´ ´ - ´ - ´ ￿ ￿ = +
- ￿ ￿ ´ ´ + ´ ´ ￿ ￿
￿ ￿
,      (10) 
  ( ) ( ) y = - = ' ' 0 B h x h x .            (11) 
 
Now, it is possible to compare leader and follower 
efforts by deriving the sign of g.(z). With g.(z)>0, the 
leader invests more than each follower. With g.(z)<0, the 
reverse applies.   
 
 
Proposition 1: With exogenous entry and no market 
sharing (￿=1), the leader invests less than each 
follower (Reinganum, 1985). But with market sharing 
(￿<1), this tendency may be reversed in case B.  
 
 
Sharing among all entrants tends to reduce their 
individual efforts in case B. A numerical example
1 
confirms this reversal: V=100, F=10, ￿=20, r=0,10, n=4 
and let  ( ) h x x = . 
  1 s =     ￿  1918   z=1974
L z = <      
0 ,5 s =     ￿  1353   z=332
L z = >      
 
1
0,25
n
s = =   ￿   644    z=43 L z = >  
 
Proposition 2: With endogenous entry and no market 
sharing (￿=1), the leader invests more than each 
follower (Etro, 2004). With market sharing (￿<1), 
                                                
1 All numerical examples are computed with Maple.   5 
this tendency remains valid in case B but may be 
reversed in case A.   
 
Losing part of the new market by the leader in case A 
tends to discourage its efforts, even if more could 
reduce the number of imitators. A numerical example
2 
confirms this reversed prediction: V=100, F=10, ￿=20, 
r=0,10 and  ( ) h x x = . 
 
  1 s =     ￿  2500   z=2025 L z = >   and  * 4 ,387 n =    
  0 ,5 s =   ￿   571   z=2025
L z = <   and   * 5,454 n =  
 
1
0 ,2
5
s = =   ￿   94   z=2025 L z = <     and   * 5,590 n =  
   
These tendencies of case A were also found in a strategic 
investment game with leaders and followers, no 
uncertainty and exogenous entry (Vandekerckhove and De 
Bondt, forthcoming). Large spillovers from a leader to 
imitators may result in them investing less than each 
follower, even though a large effort could improve their 
subsequent Stackelberg profits.  
 
 
5. Conclusion 
 
Existing theoretical and empirical work on innovative 
activities points to the need for a careful handling of 
the spillovers between participants. In leader follower 
race-settings, the incorporation of asymmetric spillovers 
can help the search for richer hypotheses to be tested in 
empirical work. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ° Corresponding author, Tel : +32 16 326904; Fax: +32 16 
326732; E-mail: Jan.Vandekerckhove@econ.kuleuven.be.  
                                                
2 Integer constraint on n is ignored here.    6 
References 
 
Czarnitzki, D. and K. Kraft, 2004, An Empirical Test of 
the Asymmetric Models on Innovative Activity: Who Invests 
more into R&D, the Incumbent or the Challenger?, Journal 
of Economic Behavior and Organization 54, 153-173   
 
Etro, F., 2004, Innovation by Leaders, The Economic 
Journal 114, 281-303. 
 
Reinganum, J., F., 1985, A Two-Stage Model of Research 
and Development with Endogenous Second-Mover Advantages, 
International Journal of Industrial Organization 3, 275-
292. 
 
Stewart, M.B., 1983, Noncooperative Oligopoly and 
Preemptive Innovation without Winner-Take-All, The 
Quarterly Journal of Economics, 681-694. 
 
Vandekerckhove, J. and R. De Bondt, 2007, Asymmetric 
Spillovers and Investment in Research and Development of 
Leaders and Followers, Economics of Innovation and New 
Technology, forthcoming. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   7 
Appendix         
INNOVATION BY LEADERS WITHOUT WINNER-TAKE-ALL 
by Raymond De Bondt and Jan Vandekerckhove 
 
The expected value of discounted profits for the leader: 
  ( ) ( )
( ) ( )
p ￿
=
￿
=
´ + ´ + -
= -
+ +
1 2
1
1
n L L j L L
j L
n j L
j
h z P h z P z
V F
r h z h z
.                        (A1) 
 
The expected value of discounted profits for each follower: 
( ) ( ) ( )
( ) ( )
￿
=
¹
￿
=
´ + ´ + ´ -
= -
+ +
1 2 3
1
1
n i i L i j i i
j
j i i
n j L
j
h z P h z P h z P z
V F
r h z h z
.                      (A2) 
 
Case A 
Combining Table 1 with (A1) and (A2) results in the following expected profits functions for the 
leader and for each entrant in case A.   8 
  ( )
( ) ( )
1
L L
L
n j L
j
h z P z
V F
r h z h z
s p
￿
=
´ ´ + -
= -
+ +
                            (A3) 
  ( ) ( )
( ) ( )
1
1 i L i
i
n j L
j
h z P h z P z
n V F
r h z h z
s
￿
=
- ￿ ￿ ´ + ´ ´ - ￿ ￿
￿ ￿ = -
+ +
                        (A4) 
 
For each of the n entrants, the first order condition needs to be satisfied,  0
i
i
V
z
¶
=
¶ .  
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
1
2
1
1
' 1 '
0
n i j L i i L i
i j
i
n j L
j
h z P r h z h z h z h z P h z P z
n V
z
r h z h z
s
￿
=
￿
=
￿ ￿ ￿ ￿ - ￿ ￿ ￿ ￿ ￿ ￿ ´ - ´ + + - ´ ´ + ´ ´ - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ¶ ￿ ￿ ￿ ￿ ￿ ￿ = =
¶ ￿ ￿
+ + ￿ ￿
￿ ￿
           (A5) 
In a symmetric equilibrium, the following applies 
( ) ( ) ( ) ( ) ( ) ( )
1
' 1 ' 0 i L L h z P r n h z h z h z h z P h z P z
n
s
f
￿ ￿ - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ = ´ - ´ + ´ + - ´ ´ + ´ ´ - = ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿
            (A6) 
The sign of  L
dz
dz  is equal to the sign of  ( ) ( ) r n h z P n z r P s ￿ ￿ + ´ ´ ´ - ´ - ´ ￿ ￿. So for s s < °, the efforts are 
strategic substitutes but for  1 s s ° < £ , the efforts are strategic complements, with 
( )
r P n z
r P n h z P
s
´ + ´
° =
´ + ´ ´ .   9 
 
The leader maximizes its profits by choosing z
L. Thus,  
 
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
2
1
' 1 '
0
L L L L L
L L
L
n j L
j
n h z
h z P r n h z h z h z h z P z
z V
z
r h z h z
s s p
￿
=
￿ ￿ ¶ ´ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ´ ´ - ´ + ´ + - + ´ ´ ´ + - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ¶ ￿ ￿ ¶ ￿ ￿ = =
¶ ￿ ￿
+ + ￿ ￿
￿ ￿
           (A7) 
or 
( ) ( ) ( ) ( )
( ) ( ) ( ) ' 1 ' 0
L L L L L L
L
n h z
h z P r n h z h z h z h z P z
z
f s s p
￿ ￿ ¶ ´ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ = ´ ´ - ´ + ´ + - + ´ ´ ´ + - = ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ¶ ￿ ￿
           (A8) 
 
Now define the following function: 
 
( ) ( ) ( ) ( ) ( ) ' 1 L A
A g x h x P r n h z h z h x P x s y s p ￿ ￿ ￿ ￿ ￿ ￿ = ´ ´ - ´ + ´ + - ´ ´ ´ + - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿                (A9) 
with  ( ) 0 L
A g z =  and  ( )
( ) ( )
'
A
L
n h z
h x
z
y
¶ ´
= +
¶  
Then,    10 
( ) ( ) ( ) ( ) 2
2
' ' ' 1 0
L
L
L
V
g x h z h z P
x
s
¶
= - ´ ´ ´ - <
¶
                         (A10) 
 
Combining (A9) and (A6) and evaluating in x=z,  
( ) g z = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
1
' 1 ' L A L h z P r n h z h z h z P z h z h z P h z P z
n
s
s y s p s
￿ ￿ - ￿ ￿ ￿ ￿ - ´ - ´ ´ + ´ + - ´ ´ ´ + - + ´ ´ ´ + ´ ´ - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿
     (A11) 
The sign of (A11) drives the comparison of z and z
L.   
 
-  Exogenous entry 
·  With  1 s = ,  ( )
( ) ( )
( ) ( )
1 ' 0 A L
n h z
g z h z P z h z
z s p p
=
¶ ´
￿ ￿ = - ´ ´ + - - ´ < ￿ ￿ ¶               (A12) 
Since z and z
L are strategic complements for  1 s = ,  0 L
z
z
¶
>
¶ , from A6.  
 
·  The sign of  ( ) A g z  for  1 s <  is unclear in general. Numerical analysis suggests that in a 
wide range of cases, the sign remains negative.  
   11 
-  Endogenous Entry 
With endogenous entry, the zero profit condition states that  0
i V =  for i=1,2,…,n. From (A4): 
  ( ) ( ) ( ) ( )
1
0 L L ZPC h z P h z P z F r n h z h z
n
s - ￿ ￿ ￿ ￿ = ´ + ´ ´ - - ´ + ´ + = ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿
               (A13) 
  Combining (A6) and (A13) yields 
    ( ) ( ) ' 1 h z P F ´ - =                             (A14) 
From (A14), z can be derived and it is clear that z is not dependent on z
L. 
 
From the Zero Profit Condition of the followers,  ( )
( ) ( )
' A L
L
n h z
h z
z
y
¶ ´
= +
¶  can be calculated, as  
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( )
2
1
'
' ' ' ' 0 '
1
( )
L
L
A L L L L
L L L L
h z P F
n h z n ZPC z n dz
h z h z h z n h z h z h z h z
ZPC n z z dz h z
P F
hz n
s
y
s
￿ ￿ - ￿ ￿ ´ ´ - ￿ ￿ ￿ ￿ ￿ ￿ ¶ ´ ¶ ¶ ¶ ￿ ￿ ￿ ￿ = + = + ´ + ´ ´ = + - ´ + = + ￿ ￿ ￿ ￿ ¶ ¶ ¶ ¶ ￿ ￿ - ￿ ￿ ´ ´ + ￿ ￿
￿ ￿
. (A15) 
Finally, A(15) should be introduced in A(11). For  1 s = ,  ( ) 0 g z > , thus  L z z > . For  1 s < , the 
sign of g(z) can be positive or negative.   
   
   12 
Case B 
Combining Table 1 with (A1) and (A2) results in the following expected profits functions for the 
leader and for each entrant in case A. 
  ( )
( ) ( )
1
L L
L
n j L
j
h z P z
V F
r h z h z
p
￿
=
´ + -
= -
+ +
                            (A16) 
 
( ) ( )
( ) ( )
1
1
1
1
n i j i
j
j i i
n j L
j
h z P h z P z
n
V F
r h z h z
s
s ￿
=
¹
￿
=
- ￿ ￿ ´ ´ + ´ ´ - ￿ ￿ - ￿ ￿
= -
+ +
                      (A17) 
 
For each of the n entrants, the first order condition needs to be satisfied, 0
i
i
V
z
¶
=
¶ .  
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
1 1
2
1
1
' 1 '
1
0
n n i j L i i j i
j j i j i
i
n j L
j
h z P r h z h z h z h z P h z P z
n
V
z
r h z h z
s
s s ￿ ￿
= =
¹
￿
=
￿ ￿
￿ ￿ - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ´ ´ - ´ + + - ´ ´ ´ + ´ ´ - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ - ￿ ￿ ￿ ￿ ￿ ￿ ¶ ￿ ￿ = =
¶ ￿ ￿
+ + ￿ ￿
￿ ￿
        (A18) 
In a symmetric equilibrium, the following applies 
( ) ( ) ( ) ( ) ( ) ' 1 ' 0 i L h z P r n h z h z h z h z P z f s ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ = ´ ´ - ´ + ´ + - ´ ´ - = ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿                 (A19)   13 
 
The leader maximizes its profits by choosing z
L. Thus,  
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
2
1
' 1 '
0
L L L L L
L L
L
n j L
j
n h z
h z P r n h z h z h z h z P z
z V
z
r h z h z
p
￿
=
￿ ￿ ¶ ´ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ´ - ´ + ´ + - + ´ ´ + - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ¶ ￿ ￿ ¶ ￿ ￿ = =
¶ ￿ ￿
+ + ￿ ￿
￿ ￿
             (A20) 
or 
( ) ( ) ( ) ( )
( ) ( ) ( ) ' 1 ' 0 L L L L L L
L
n h z
h z P r n h z h z h z h z P z
z
f p
￿ ￿ ¶ ´ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ = ´ - ´ + ´ + - + ´ ´ + - = ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ¶ ￿ ￿
             (A21) 
 
Now define the following function: 
( ) ( ) ( ) ( ) ( ) ' 1 L B
B g x h x P r n h z h z h x P x y p ￿ ￿ ￿ ￿ ￿ ￿ = ´ - ´ + ´ + - ´ ´ + - ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿                  (A22) 
with  ( ) 0 L
B g z =  and  ( )
( ) ( )
'
B
L
n h z
h x
z
y
¶ ´
= +
¶  
Then,  
( ) ( ) ( ) ( ) 2
2
' ' ' 1 0
L
L
B L
V
g x h z h z P
x
¶
= - ´ ´ - <
¶
                          (A23) 
   14 
The sign of (A22) drives the comparison of z and z
L.   
-  Exogenous entry 
·  With  1 s = ,  ( ) ( )
1 0 B
B g z h z P z
s y p
=
￿ ￿ = - ´ ´ + - < ￿ ￿ , by which  L z z < . 
·  With  1 s < , it is possible that  ( ) 0 B g z > , by which  L F z z > . The example in the text is a proof 
of this.  
-  Endogenous entry 
With endogenous entry, the zero profit condition states that  0 i V =  for i=1,2,…,n. From (A17) 
  ( ) ( ) ( ) 0 L ZPC h z P z F r n h z h z ￿ ￿ = ´ + - - ´ + ´ + = ￿ ￿ ￿ ￿                     (A24) 
  Combining (A19) and (A24) yields 
    ( ) ( ) ' 1 h z P F s ´ ´ - =                            (A25) 
From (A26), z can be derived and it is clear that z is not dependent on z
L. 
 
From the Zero Profit Condition of the followers,  ( )
( ) ( )
' B L
L
n h z
h z
z
y
¶ ´
= +
¶  can be calculated, as    15 
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
'
' ' ' ' 0 ' 0
L L
B L L L L
L L L L
h z F n h z ZPC z n dz
h z h z h z n h z h z h z h z h z
ZPC n z z dz h z F
y
￿ ￿ - ´ ￿ ￿ ¶ ´ ¶ ¶ ¶ ￿ ￿ = + = + ´ + ´ ´ = + - ´ + = + ´ - = ￿ ￿ ￿ ￿ ￿ ￿ ¶ ¶ ¶ ¶ - ´ ￿ ￿ ￿ ￿ ￿ ￿
. (A26) 
Consequently,  
( ) ( ) ( ) ( ) ' 1 0 L
B g z h z P r n h z h z ￿ ￿ ￿ ￿ = ´ - ´ + ´ + > ￿ ￿ ￿ ￿ ￿ ￿ , by which z
L>z for 
1
1
n
s £ £ .          (A27) 
 

Innovation by leaders without winner-take-all.

https://lirias.kuleuven.be/bitstream/123456789/120446/1/MSI_0709.pdf

Innovation by leaders without winner-take-all.

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics