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+ ∙∙ + xr^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 2subgames. 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= xfor all t. If k nx, then
v (x, x1) ∕x= v (x1,x2) ∕xιy = ∙∙ = V (x, xS)x^ x2l = - 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= xand x= xfor t = 2, ∙∙, n. If k x+ (n 1) x,
then from (S)
v (x, xf) xl- = 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,xe) = dv (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, xe) /dxv


/(k tx)

(n t)


0) x


(19)


v (X, xe) /dx∂v ((k(~ (t +^1ζap, 0) x0                 (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.

28



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