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



More intriguing information

1. Subduing High Inflation in Romania. How to Better Monetary and Exchange Rate Mechanisms?
2. Business Networks and Performance: A Spatial Approach
3. Tissue Tracking Imaging for Identifying the Origin of Idiopathic Ventricular Arrhythmias: A New Role of Cardiac Ultrasound in Electrophysiology
4. The Advantage of Cooperatives under Asymmetric Cost Information
5. Multifunctionality of Agriculture: An Inquiry Into the Complementarity Between Landscape Preservation and Food Security
6. The Impact of Minimum Wages on Wage Inequality and Employment in the Formal and Informal Sector in Costa Rica
7. The Economic Value of Basin Protection to Improve the Quality and Reliability of Potable Water Supply: Some Evidence from Ecuador
8. The name is absent
9. Foreign direct investment in the Indian telecommunications sector
10. A Rare Case Of Fallopian Tube Cancer