f(x) = kx2 f(y) = ky2 , k>0. A detailed study of the second stage equilib-
rium of this game can be found in Gabszewicz and Thisse (1979) and Shaked
and Sutton (1982). The subgame perfect equilibrium of the whole game can
be obtained by backward induction and first stage overall payoffs for firm 1 are
given as follows:
F (x, y)=
U(x,y) = 4(χtχ-y)y) - kx2 if x ≥ y
(14)
L(x, y) = y{yyv-x) - kx2 if x < У
Now we can check whether assumptions of Theorem 4.1 are fulfilled. Given
that U and L are differentiable we can check supermodularity (Assumption B1)
by recurring to Topkis’s Characterization Theorem. Consider that U1 (x, x)=
9 — 2x and Li (x, x) = — 1 — 2x, so Ui (x, x) > Li (x, x) which verifies B2. To
check B3 we compute:
U.,(~F,(ιι∖ 4} — — 4r2 (ιΛ 2rι(y)+y rπ (r (r,.∖ ,л _ _r2 ( A 2y+rι(y)
U2 (r1 (y) ,y)= 4ri (y) (4rι(y)-y)3 L2 (ri (y) ,y) = r1 (y) (-4y+rι(y))3
1 (15)
Since ri (y) > y > ri (y) the condition B3 holds. We know, hence, that the
reaction curves are upward sloping except for a downward jump at a point d.
Since U(0, 0) is not defined we cannot check the first part of B4 directly, we
must compute limx→o Ui (x, x). We obtain limx→o Ui (x,x) = 9 > 0 which
means that increasing quality investment at point (0, 0) is profitable for both
firms. Point (0, 0) is thus ruled out as a pure strategy Nash equilibrium of
the game. As for the second part of B4, it is easily verified by the fact that
Li (c, c) = — (9 + 2kc) < 0. From Lemma 4.1 we have that if there is an
equilibrium, it cannot be symmetric in [0, c]2.
To apply Theorem 4.1 we must also check, whether B 5 is verified. To this
end, we must find the point d where the reaction curve has a jump. If the
reaction curve is given implicitly, there is no algorithm to find this point d,
however it is possible to find it through numerical methods. In the case of the
model by Aoki and Prusa (1996), the point d is equal to . It is possible to
compute that Ui (x, 0) = 4 — 2kx ≥ 0 for x > d and that for sufficiently big c,
Li (x, c) = limy→∞ Li(x, y) = —2kx + iɪʒ < 0 for x > d. So the assumptions
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