Strategic Investment and Market Integration

From (S), we have x'.™ = x%-_-ι > x ⇒ k_n-2 > 3x. If (D) holds with equality, i.e. x = x^m, then
the LHS of equality [14] equals to zero and the equality is satished if and only if k_n-2 — x =
2χ^m ⇒ k_n-1 = x + 2x^m. Work in the same way inductively to period 1. In period 1, we have
x™ = x^-1 = ■■ = '∙^l'-2 > x ⇒ k > nx and as (D) holds with equality k_n-ι = x + (n — 1) x^m. The
entry deterring capacity k1 is implicitly dehned by

— (x, β¹ (T)) = — ( ^kL^-Ξ. , oʌ for t = 2,..,n                (16)

∂x^ ^{V ,f ( )J} ∂xψ \— — 1 ^, 1             ^{, ,                            ( )}

and we conclude that k¹ ∈ (nx, x + (n — 1) T^m].

Step 5. In a subgame without entry

∂v / _m ∂V k k — xψ ∖ _rλ

dx™  '^{1 ,o) —} ∂W ( n — 1 ^{, 0}J=^{0                        (17)}

and x™ = k∕n < x^m for all t = 1,..., n. Hence, the entire capacity will be used in an equilibrium
without entry. No capacity is left idle.

Step 6. Working backward to period 0, the incumbent would install k¹ to deter entry in all markets.

27

<<   25   26   27   28   29   30   31   >>

More intriguing information