The name is absent

Example 4.- Suppose that there are three firms and that in a (sufficiently
large) neighborhood of a N.E. the relevant functions read p = a^, - x, C = cx^
2

- d/2 x - t x with a’ > c, d > O, d - 2 < O (so the second order condition
ι ι ι

holds), d - 1 > O, and C = c^,x, with a^, > c^,, i = 2, 3. Let a ≡ a^,- c and
i                 i

let a ≡ a^, - c^,. Profit maximization implies that x= (x - a ~ t^)∕(d - 1) and

x = a - x, i = 2, 3. Solving the system we get x = (2a (d-l)-a-t)/

i                                                                                                                                                               ι

(3(d - 1) - 1). If, for instance, a = 10, d = 1.5, t= 5 and a = 1 we have

that x* = 8, x* = 4, x* = 2. But if t = 5.5, x* = 7, x* = 1, x* = 3.

I    i       1        Ii

For the next Proposition we will need an additional assumption. This
assumption plus A.4 implies that a variation in t. affects both marginal and
total payoff in the same direction.

Assumption 5: U ( ) is increasing on t .
------------------------------------- ɪ                                                      i

2

Proposition 5: If all payoff functions are t? and A. 2-5 hold, an increase in

t, a) increases the payoff of i and b) decreases the payoff of any other
i

player

Proof: First, it is easy -but tedious- to show that all variables are

continuously differentiable functions of t in a neighborhood of equilibrium,
i

since assumption 2 implies that the Jacobian matrix of T ( ) has a non

vanishing determinant. Then, taking into account the first order conditions
for player j≠i, we have that

dU∕dt= ∂U ( )∕∂x о (dx∕dt - dx ∕dt )
Jij         i j i

22

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