The name is absent

The reader may wonder if,

under the aggregation axiom,

substitution alone may be sufficient to yield well defined answers to our
comparative statics questions. The following Theorem looks for structural
properties of best reply functions under these two assumptions and finds a
negative result. First, let us define x ≡ У x .

-i .ɪɪ. J

J≠ι

Theorem 1: Let x = f (x ) I = 1,..., n be a collection of S^,° functions
------------------------ ɪ i -i

defined, on a compact set and such that f ( ) is strictly decreasing ∖∕i. Then,

i

a) Vi, 3 U (x ,x), U ( ) ∈ δ^,1_j, concave on x such that
iιi         i

f (x ) ≡ arg. max U (a, a+x ), Vx
i-i        a∈Si      -i     -i

i

Moreover, U ( ) can be taken to be decreasing on x (i.e., fulfilling A.3)
i

b) ∀ , 3 a ifɪ cost function C (x ) and a linear inverse demand function
i                                                                     i i

p = A-x such that

f (x ) ≡ arg. max (A - b - x )b - C (b) Vx
i -i               b ∈ S                   - i           i            -i

i

Proof: a) First notice that f( ) is invertible. Also, f.¹( ) is integrable

since f ¹( ) e (by the continuity of f ( ), see Bartle (1976), p. 156), and
i                                                                                        i

it is bounded (see Bartle (1976) p. 427). Let q (x ) be the primitive of
i i

-1                                              2

f (x ). Define U ≡ q (x ) + x - x x. Notice that U is decreasing on x.
ii   iiiii     i

Then we have that

∂U,

---— = f ¹(x ) + 2x-x-x≡f ¹(x ) - x = 0.
_                i i             i i                 i i -i

∂X
i

and since f ¹( ) is strictly decreasing U is concave on x, so the second
i                                                                                   i                                                   i

order condition of payoff maximization is satisfied and thus, a) holds.

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