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