Strategic Investment and Market Integration

Appendix A. Sequential Entry

Proof. Step 1. Start in period n. Let the remaining capacity be k_n = k_n-1 — ^x^_-↑ ■ There are
two subgames; either player n enters or stays out of market n. In the subgame with entry, the
unique Nash equilibrium is {xt^ (k_n), -r^}, where xt^ (k_n) = min {k_n,x™} and "x^r_n = β^e (xtβ (k_n)).
If player n decides to stay out of market n, we have the following limit Nash equilibrium [x™', θ},
where ^ym (x™, θ) = 0. The unique Nash equilibrium in the subgame with no entry is {x)(^λ (k), 0},
where x™ (k_n) = min {k_n,x™}. From S, it follows that x™ > x™.

Step 2. Player n would enter if k_n < x and stay out as long as k_n ≥ x. To deter entry, player m
would need k_n ≥ x. Now, assume that enough unused capacity remains to deter entry. Rewrite the
equilibrium output of player m in period n as a function of k_n-ι and x%-ι, i.e. x™ (k_n-ι, x%-ι ) =
min {kn-i — x^rβ_-1,χ^m}.

Step 3. Working backwards to period n — 1, we have two subgames; either player n — 1 enters or
stays out of market n — 1. First, capacity k_n-ι would ensure a successful commitment by player
m in market n — 1 to an output x™—i, if and only if:

°v                   °x™   °V

-----(T™ ₁,β¹ (T™ ₁)) + —ɪ ∙ -— (x™ (k,T™ ₁), 0) ≥ 0             (13)
я„,™   ∖ n^-1^,    ∖ ^n- 1)) а^т Ягг-т ∖ ⁿ V ^{, n-1} ) ^, ) —                    V >

°^xn-1                     °^xn-1 °^xn

Now, ∂χm = —1 if k ≤ x™ + τ^m and g∂⅛ (k, 2:^-1 ) = 0 if k > x™ + x^rβ^. To deter entry, player
m has to commit to T in the subgame with entry. The following inequality must be satished:

°V               °V

-°^(^x, в™^(T)) = °   °^c, 0)                        (1⁴)

If xn-1 > 0, it follows from (S) that x^rβ > x ⇒ k_n-1 > 2T. If (D) holds with equality, i.e.

T = x™, then the LHS of equality [14] is equal to zero and the equality is satished if and only if
,         ~   =rn , ,         ~ , =rn

k_n-1 — x = x ⇒ k_n-1 = x + x .

Step ^. Working backwards to period n — 2, we have two subgames; either player n — 2 enters or
stays out of market n — 2. First, capacity k_n-2 would deter entry if:

°v                ° V

∂x^- (^t, в™ ^(T)) = ∂x^

°^xn-2              °^xn-1

°          °      °V

(x™.,, 0 =      (x™, 0)

∖ ⁿ- 1 ’ /     rn_rm × ⁿ ’ '

^x >x ^n

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