b) Let p(x) ≡ A - x and C(x ) ≡ Ax - χ2 - q (x ) + B, where q ( ) is as
i i i i i i i
defined in part a) above. Since x is defined on a compact set, B can be taken
i
large enough such that C(x ) ≥ O, Vx . Also, taking A large enough, the
i i
marginal cost is positive. Then,
2 2
π ≡ p(x)x - C(x ) = (A-x)x -Ax - x - q(x)-B = q(x) + x - xx - B
lli i i i i i i i i i i i
which is identical to the utility function constructed in part a) above.
The main consequence of Theorem 1 is that in games in which both the
aggregation axiom and the strategic substitution assumption hold, that the
best reply functions depend on the sum of strategies of the other players and
that they are decreasing exhaust all the properties of best reply functions.
Thus, they are, up to some extent, arbitrary (this result may be regarded as
analogous to the lack of structural properties of excess demand functions in
General Equilibrium, see Sonnenschein and Shafer (1982) but in our case the
root of the problem is not on the aggregation side). Even if payoff functions
are restricted to be profit functions, no structural property beyond those
quoted above can be found!.
As an easy corollary of Theorem 1 we have that a) the equilibrium set of
strategies is arbitrary and b) comparative statics will not yield definitive
answers. Both points can be easily seen in the case of two players by
constructing best reply mappings which intersect at any given set of points
and by considering shifts of these curves and comparing non adjacent
equilibria. Thus, we are lead to conclude that in general, we need additional
properties to those quoted before in order to tackle comparative statics. As
we will see our A. 2 will be sufficient for this job.
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