III.- THE EFFECTS OF ENTRY
In this Section we will study the effects of an increase in the number of
players (see Bresnahan and Reiss (1991) and the references there for the
empirical evidence in oligopolistic markets). In order to save notation let
y ≡ x (n+l). Also, let us denote by x(n), x (n) and U (n) the equilibrium
n+l i i
values of x, x and U ina game with n players.
Proposition 1 : Under A.l-2 we have that
a) x(n) ≤ x(n+l), x (n) ≥ x (n+l) MieN and
i i
b) if y > 0 the above inequalities are strict.
Proof: We first notice that if x(n) ≥ x(n+l) and x(n) > 0, xfn+l) = 0 is
impossible since T (x (n), x(n)) ≥ T (0, x(n+l)) ≥ T (0, x(n)) would
i i i i
(2)
contradict that T ( ) is strictly decreasing on x . Take any i ∈ N c∖ N+l (if
i i
i g N+l, xfn) > x(n+l) = 0). In both N.E. first order conditions hold so
(1) Tfxfn), x(n)) = Tfxfn+1), x(n+l)).
Therefore because A.2 we have only two possibilities:
I- x(n+l) ≤ x(n) and xfn+l) ≥ xfn), with a strict inequality or
II.- x(n+l) ≥ x(n) and x (n+l) ≤ x (n).
i i
(2) A similar argument shows that if x(n) ≤ x(n+l) and x(n) = 0, then x(n+l) =
0, so the second inequality in a) in Proposition 1 holds V i ∈ I.
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