Strategic Investment and Market Integration

is denoted β^e (x™). The best reply functions β^m (x1) and β^e (x™) are implicitly defined by

If a potential competitor decides to enter in period 1, this gives the following Nash equilibrium,
when the capacity constraint is non-binding for the incumbent: {x™,xe}, where x™ = β^m (ʃf),
Xi = β^e (xT)∙ If к ≤ x™, the incumbent will use the entire capacity, but with к > x™ some
capacity will be left idle. The unique Nash equilibrium in the subgame with entry is {x™ (к), x1}
where xψ^, (к) = min{k,x™} and x'1 = β^e (x↑^l (к)).

In the second subgame in the second stage, with no entry, we obtain the following Nash equilib-
rium, when the capacity constraint is non-binding for the incumbent: {x™, 0}, where x™ = β^m (0).
Thus, x™ is the monopoly level the incumbent would choose, if the cost of capacity was sunk and
capacity did not restrict output.

When the firms compete in strategic substitutes, the potential entrant’s profit is decreasing in
the incumbent’s output. However, the incumbent does not choose an output above the limit x™,
if the potential competitor enters the local market. Thus, under condition (S), it is a necessary
condition for entry deterrence that the profit of the potential entrant is non-positive in a Nash
equilibrium with a non-binding capacity restriction for the incumbent. This condition will be
denoted D:

(ŋ) V (x1, xβ} — cxf — A ≤ 0

If the necessary deterrence condition D is satisfied, condition S is a sufficient condition for entry
deterrence. However, it can easily be shown that S is not a necessary condition for the result. In
particular, the result can hold, even if the strategic variables are strategic complements.

If D is satisfied and player t would earn a positive profit as a monopoly it follows from the
Theorem of Intermediate Values that the profit of the entrant must be equal to zero at some
positive level of output by the incumbent. This deterrence level will be denoted x and defined:

π (β^e (x), x) — cβ^e (x) — A = 0

Thus, if the established firm successfully commits to an output x, it deters entry. It is also

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