assumed that x is above the output level of a natural monopoly. In other words, the entry-
deterring incumbent in our model is operating beyond the scale of operation it would choose, if it
did not face potential entry.
Next, three results from the Hrst version of the multi-market game, Γ^, can be shown. First, D
is a sufficient condition on entry deterrence in the multi-market game. Second, if Hrms compete in
strategic substitutes, then D is not only a sufficient, but a necessary, condition for entry deterrence.
Third, if both conditions S and D are satisHed, the incumbent installs strictly more than n ■ x to
deter entry in Γ^.
If D is satisHed the local entrant does not earn a positive proHt in {x"trι∙ xf}, and would thus stay
out of the local market. To see that D is a sufficient condition for entry deterrence, assume that
the incumbent has installed more capacity in period 0 than he will ever use. Thus, every market
can be treated independently and the unique Nash equilibrium in every market t is {x'"λ∙X∕''}, and
entry deterrence is thus possible.
Proposition 1 If D is satisfied, then entry deterrence is possible in Γ^.
Proof. (D ⇒ entry deterrence is possible). Let the pre-commitment capacity be very large. The
capacity constraint is not binding in any subgame. The objective of the incumbent is to maximize
its proHt with respect to x™, for all t,
∂V (x™, xf)
0 ∀t.
(5)
∂xfl
This problem is additively independent and each market can be considered as a separate one-
market game Γ1. If the capacity constraint is not binding, the unique Nash equilibrium with entry
is ∖xr∣n.xl}, where x™ = β0 (xt), xt = βt (xlfif. Since v (xt,x7fi) — cxt — A ≤ 0, player t will choose
to stay out and monopoly prevails. ■
Next, we will show that, the deterrence condition (D) is not only a sufficient, but also a necessary
condition on entry deterrence, if Hrms compete in strategic substitutes. Strategic substitutes (S)
imply that the proHt of a potential entrant is monotonically decreasing in the incumbent output.
If к does not restrict output, then {x^rXt} is the unique Nash equilibrium with entry in market
t. Furthermore, x™ is the highest output the incumbent will select with any capacity к. Hence,
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