Strategic Investment and Market Integration

if the potential competitor earns a positive profit in {x—,x^t}, the same will hold in any Nash
equilibrium in the post-entry game. Thus it enters market t and entry deterrence is not possible.

Proposition 2 If condition S is satisfied and condition D is violated, then entry deterrence is not
possible in Γ^.

Proof. (S and ^ D ⇒^ entry deterrence). First, note that x— is player m’s highest output
level in a subgame with entry in market t. From (S), π (xf, x—) is monotonically decreasing in x—
and reaches its minimum at x—. If ^ D, i.e. v χ^t,x— — cx^t — A > 0, player t could ensure a
positive profit, if entering market t. ■

After these two qualitative results, a more precise result can be established, characterizing
the disadvantage of multi-market competition on entry-deterrence. If firms compete in strategic
substitutes and the necessary deterrence condition is satisfied, the incumbent must install к > nX
to deter entry in the п-market game Γ^.

Consider for instance the two-market game. Why is twice the deterrence level, X, not enough
to deter entry in two markets? The main reason is that if one potential competitor enters and the
other stays out, the incumbent has an incentive to redistribute capacity to the monopoly market.

In the last period, the remaining capacity is к — x—. If condition D is satisfied, к — x— ≥ X
will deter entry. Working backwards to period 1, there are two subgames. If player 1 stays out,
the incumbent will split the capacity equally in both markets. If the potential competitor enters,
the marginal incentive to use capacity in market 1 and 2 must be equal:

∂π               ∂π

_-~- (x—, β* (x—)) = --- (к — x—, 0)                         (6)

xi∙j                      x2p

It follows from strategic substitutes that к — x— > x—. Thus, if к = 2X, then x— < X and entry is
not deterred in the first market. More specifically,

Proposition 3 If D and S are satisfied in the first version of the п-market game, Γ∖, then the
τ1         о       =0

multi-market incumbent installs capacity nx < к_п ≤ x + (п — 1) x to deter entry.

Proof. Appendix A ■

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