Strategic Investment and Market Integration

Appendix B. Simultaneous Entry

This proof is valid for the main result in the second and third versions of the multi-market game.

Proof. Step 1. Begin in stage two. The objective of player m in the second stage is to solve the
following program:

max υ (x™, xf ) + V (x™, xf) + ∙∙ + v (ʃ^, x”)
_o f  ™n ™7l I I ™n < ‰

s.t. X1 + Xf + ∙∙ + X^ ≤ rv

If x™ + x™ + ∙∙ + x^r^ < k, then ∂v (x^^,_lⁿ, xf^) ∕∂x™ = 0 for t = 1, ..., n. If x™ + x™ = k, then
∂v (x™, x↑) ∕∂x™ = ∂v (x™, xf) ∕dx™ = ∙∙ = ∂V (x™, x”) ∕∂x™.

Step 2. In the last stage there are 2” subgames. First, if entry does not occur in any market and
k > nx™, then ∂v (x^^,∣ⁿ, 0) ∕∂x^^ra = 0 for all t = 1, ..., n ⇒ x™ = x™ for all t. If k ≤ nx™, then
∂v (x™, x1) ∕∂x™ = ∂v (x™¹,x2) ∕∂x^ιy = ∙∙ = ∂V (x™, xS) ∕∂x^ ⇒ x2^l = ^- for all t.
∖l^l∕/ I         ∖ c_ ¹ C_ / I ”               \”^”//    ”       l ”

Step 3. Second, if one player enters (w.l.o.g. player 1) and k > x™+(n — 1) x™, then ∂v (x™, xf) ∕∂x¹y =
0 and ∂V (x™, 0) ∕∂x^ = 0 ⇒ x™ = x™ and x™ = x™ for t = 2, ∙∙, n. If k ≤ x™ + (n — 1) x™,
then from (S) ∂v (x™, xf) ∕∂x^l-∖ = ∂v (x™, 0) ∕∂x^ for t = 2, ∙∙∙,n ⇒ x™ < k∕n and x™ > k∕n. To
deter the entry of a single entrant while n — 1 players stays out, the incumbent must install

∂V (x™,x^e) = ^dv (⅛≡d, ⁰)
dx™         dx™                                ^{l )}

and from (S) k > nx.

Step X Next, if capacity k deters the entry of a single entrant, k deters the entry of more than

one player, which is shown with induction. Assume k deters the entry of t players. Then
where entry occurs in i and no entry occurs in market j. If t +1 players enter, deterrence is credible
if

∂v (X™, x^e) /dx™ — ∂v (^(k(~ ^(t +^¹ζ^ap, 0) ∕∂x™ > 0                 (20)

where entry occurs in i and no entry occurs in market j. The last inequality holds as long as
k > nx. Hence, we have shown that if capacity k deters the entry of a single entrant, then k deters
the entry of more than one entrant.

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