and Proposition 4 and A. 3 imply b) above. In the case of player i we have
that
dU ∕dt = ∂U ( )∕∂x о (dx∕dt - dx ∕dt )+ ∂U ( )∕∂t
i i i i i i i i
and since the strategy of all competitors has decreased and A. 5 we obtain a)
above.*
The next example shows the necessity of A. 2 for Proposition 5 to hold
Example 5.- Suppose that the market is as in example 4. Then it is easily
calculated that if t = 5, U*= 4 and U* = 4, i = 2, 3. But if t = 5.5, U* = 0
11 i 1 1
and U*= 9, i = 2, 3.
i
We will end this Section by studying the effects of a generalized shock.
Proposition 6: Under A. 1, 2 and 4 an increase in t increases x
Proof: First, by analogous reasoning to Proposition 1 it can be shown that x
can not be constant. And if x decreases all x must increase. Contradiction.*
i
The effect of t on individual strategies and payoffs in equilibrium
depends on how payoff functions are affected (see Dixit (1986) and Quirmbach
(1988)). This means that, in the Cournot model, a technological improvement in
costs might decrease the output and profits of the most efficient firm (see
example 6). Finally without A. 2 Proposition 6 does not hold (see example 7).
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