Competition In or For the Field: Which is Better

A Proof of Proposition 2

Differentiating both sides of π ◦ π ¹ (p) = p leads to:

Hence:

(9)                           ( S ◦^{π -1}) 0 ( p ) = S 0 (^{π -1(} p ))(^{π -1}) 0 ( p ) = S 0(^π₁ ((^p))

π⁰(π^-1(p))

Differentiating both sides of (8) with respect to p and using (9) leads to:

S 00 π0 - S 0 π 00
(∏ 0 )³

where all terms on the r.h.s. are evaluated at π^-1 (p). Since π0 > O this implies that S ◦ π^-1 is concave if
and only if S00π0 < S0π00. And since S0 < 0 and π0 > 0 we conclude that π00/π0 < S00/S0 is necessary and
sufficient for concavity of S ◦ π^-1. The result now follows.

B Proof of Lemma 2

To prove (i) and (ii) totally differentiate the identity q ≡ D[P(q)] with respect to q to obtain

1 = D⁰P⁰ ,

from which (i) follows. Next, totally differentiate P0(q) ≡ D0[P(_q)] with respect to q to obtain

D⁰⁰
P = ^-(∣>^,)2P ^,

from which (ii) follows by substituting D0 for P0. The proof of (iii) and (iv) is analogous and we omit it.

C Proof of Lemma 3

Proof: Sufficiency: use (i) and (ii) in Lemma 1 to substitute for P⁰ and P⁰⁰. Necessity: use (iii) and (iv) to
substitute for D, D⁰ and D⁰⁰ .

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