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

p i = a − q i − b X j6=i
1.2 Increasing Marginal Costs and Product Differentiation 19
q j (1.25)
where b ∈ (0, 1] is an index of substitutability between goods. Of course, for
b =0goods are perfectly independent and each firmsellsitsowngoodas
a pure monopolist, while for b =1we are back to the case of homogeneous
goods. In this more general framework the profit functionforfirm i is:
π i = q i
⎛
⎝a − q i − b
nX
j=1,j6=i
q j
⎞
⎠ − cq i − F (1.26)
The four main equilibria can be derived as usual. In particular a Nash equilibrium
would generate the individual output:
a − c
q(n) =
2+b(n − 1)
for each firm. In the Marshall equilibrium each firm would produce:
(1.27)
q = √ F (1.28)
with a number of firms:
n =1+ a − c
b √ F − 2 b
Under Stackelberg competition, the leader produces:
(a − c)(2 − b)
q L =
2
and each follower produces:
(1.29)
(a − c)[2 − b(2 − b)]
q(n) = (1.30)
2[2 + b(n − 2)]
Finally, consider Stackelberg competition with endogenous entry. As long as
substitutability between goods is limited enough (b is small) there are entrants
producing q(q L ,n)=(a − bq L − c)/[2 + b(n − 2)]. Setting their profits equal
to zero, the endogenous number of firms results in:
n =2+ a − bq L − c
b √ − 2
F b
implying once again a constant production:
q = √ F (1.31)
for each follower. Plugging everything into the profit function of the leader,
we have: