Competition, Innovation, and Antitrust. A Theory of Market ... - Intertic

1.4 A Simple Model of Competition for the Market 27
Consider now a Stackelberg equilibrium. As already noticed, when the
investment by one firm is increased, the other firms have incentives to invest
less: then in a Stackelberg equilibrium the leader exploits its first mover advantage
by investing more than the followers, so as to reduce their investment
and increase its relative probability of winning. For instance, in a Stackelberg
duopoly the leader invests z L = V (1 − V )/(1 − 2V 2 ) and the follower invests
z = V (1 − V − V 2 )/(1 − 2V 2 ).
In a Stackelberg equilibrium with endogenous entry, as long as the investment
of the leader z L is small enough to allow entry of at least one firm, the
first order conditions and the free entry conditions are:
(1 − z) n−2 (1 − z L )V = z, z(1 − z) n−2 (1 − z L )V = z 2 /2+F
which deliver the same investment choice by each entrant as in the Marshall
equilibrium, z = √ 2F , and the number or firms:
h
log (1 − z L )V/ √ i
2F
n(z L )=2+ h
log 1/(1 − √ i
2F )
Putting these two equations together and substituting in the profit function
of the leader, we would have:
π L = z L (1 − z) n−1 V − z2 L
2 − F =
= z √ ³
L 2F 1 − √ ´
2F − z2 L
1 − z L 2 − F (1.41)
which has not an interior optimum: indeed, it is always optimal for the leader
to deter entry investing enough. This requires a slightly higher investment
than the one for which thehequilibrium number of firms would be n =2.
Since n(z L )=2requires log (1 − z L )V/ √ i
2F =0, we can conclude that the
leader will invest:
√
2F
¯z L =1−
(1.42)
V
which is increasing in the value of innovations and decreasing in their fixed
cost. Therefore, in a contest with a leader and free entry of participants, the
leader invests enough to deter investment by the other firms and is the only
possible winner of the contest.
1.4.1 The Arrow’s Paradox
Until now we investigated a form of competition for the market where all firms
were at the same level. Often times, competition for the market is between an