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We now establish the contraction property of the mapping. Using symmetry across agents⎛ ⎞IbAΣ e A ′ b ′ = b i (I · · · I) AΣ e A ′ ⎜ .⎟⎝ ⎠ bi′ = b i N(I + p)b i′ .INext note that using Lemma 3(I + γ i )(µ ′ (I + γ i ) + (I + γ i′ )µ) −1 = 1 2 µ−1 = 1 2 (bAΣ eV ′ ) −1 (bAΣ e A ′ b ′ + Σ u ),where the last equality follows from the definition of µ. Substituting for bAΣ e A ′ V ′ and bAΣ e A ′ b ′into µ −1 yields12 µ−1 = 1 2 (Nbi (I + p)) −1 (Nb i (I + p)b i′ + Σ u ).Now substitute for 1 2 µ−1 into the first-order condition for agent i, equation (20) to obtain(I + p)b i′ = pb i′ 1 2 (I + γi ) −1 + 1 2 (I + p)(Nbi (I + p)) −1 (Nb i (I + p)b i′ + Σ u )⇒ b i′ = (I + p) −1 pb i′ 1 2 (I + γi ) −1 + 1 2 (Nbi (I + p)) −1 (Nb i (I + p)b i′ + Σ u ).Multiplying both sides by Nb i (I + p) yieldsNb i (I + p)b i′ = Nb i (I + p)(I + p) −1 pb i′ 1 2 (I + γi ) −1 + 1 2 (Nbi (I + p)b i′ + Σ u )Next subtract 1 2 (Nbi (I + p)b i′from both sides to obtain12 Nbi (I + p)b i′ = Nb i (I + p)(I + p) −1 pb i′ 1 2 (I + γi ) −1 + 1 2 Σ u.Since p is diagonal and b i′is symmetric and countersymmetric, then again by Lemma 3 we cancommute (I + p) −1 pb i′to b i′ (I + p) −1 p and divide by N 2to obtainb i (I + p)b i′ = b i (I + p)b i′ (I + p) −1 p(I + γ i ) −1 + 1 N Σ u. (21)Defining the 2 × 2 matrix Q ≡ b i (I + p)b i′ , we can write equation (21) asQ = Q(I + p) −1 p(I + γ i ) −1 + 1 N Σ u.From lemma 2‖(I + γ i ) −1 ‖ < 2.35
It is straightforward to establish that the maximum singular value of (I + p) −1 p isθ1−θ . For θ < 1 3 ,we then haveθ1−θ < 1 2 , so that ‖(I + p) −1 p(I + γ i ) −1 ‖ < 1,implying that the mapping of Q is a contraction. We recover the fixed point of b i by finding theCholesky factor of the fixed point of Q and multiplying that factor by the inverse of the Choleskyfactor of I + p. The Cholesky factor of I + p is⎛⎜⎝ 1θ0 (1 − θ 2 ) 1 2⎞⎟⎠ .Proof of Proposition 4.Matrices that are both symmetric and countersymmetric commutewith each other. Exploiting that γ i is symmetric and countersymmetric and identical across agentswe multiply the first-order condition for b i , equation (10), by b i and add across i to obtainbAΣ e A ′ b ′ Γ ′ λ ′ (I + γ i ) + 2b i A i Σ e A i′ b i′ λΓ + bAΣ e bA ′ b ′ γ i′ λΓ = bAΣ e V ′ (I + γ i ). (22)Using the definition of Ψ in (9), we next substitute Ψ − Σ u in the first term to rewrite (22) as(Ψ − Σ u )Γ ′ λ ′ (I + γ i ) + 2b i A i Σ e A i′ b i′ λΓ + bAΣ e A ′ b ′ γ i′ λΓ = bAΣ e V ′ (I + γ i ) = ΨΓ ′ λ ′ (I + γ i ),where the last equality follows from the first-order condition for λ, equation (12). Subtracting(Ψ − Σ u )Γ ′ λ ′ (I + γ i ) from both sides yields2b i A i Σ e A i′ b i′ λΓ + bAΣ e A ′ b ′ γ i′ λΓ = Σ u Γ ′ λ ′ (I + γ i ) = Σ u (I + γ i )Γλ,where the final equality follows from the symmetric-countersymmetric structure of Γ ′ λ ′ and I + γ i ,which allows us first to transpose and then commute. Dividing by λΓ then yields2b i A i Σ e A i′ b i′ + bAΣ e A ′ b ′ γ i′ = Σ u (I + γ i ). (23)We now use first-order condition for γ, equation (11), to simplify (23). Multiplying both sides of(11) by Ψ = bAΣ e A ′ b ′ + Σ u and re-arranging yieldsbAΣ e A ′ b ′ γ i′ = − ( Σ u γ i′ + bAΣ e A i′ b i′) . (24)36
Page 1 and 2: Cross-Asset Speculation in Stock Ma
Page 3: ship model of Kyle (1985) and Admat
Page 6 and 7: We next set out the model and analy
Page 8 and 9: of net order flows. Specifically, w
Page 10 and 11: and defining γ i ≡ B i (I −
Page 12 and 13: (from Proposition 1) speculator i
Page 14 and 15: Unobservable prices. This theoretic
Page 16 and 17: 3, speculative trading intensities
Page 18 and 19: the off-diagonal term in Γ = Γ
Page 20 and 21: Substituting for λ 11 − c = λ 1
Page 22 and 23: isolates how the division of privat
Page 24 and 25: mation is highly correlated. Profit
Page 26 and 27: The increased competition between s
Page 28 and 29: of the less volatile asset’s valu
Page 30 and 31: V AppendixProof of Proposition 1. U
Page 32 and 33: under the trace yields{0 = tr= tr
Page 34 and 35: where once more we commute under th
Page 38 and 39: We substitute (24) for bAΣ e A ′
Page 40 and 41: We have⎛ ⎞V Σ e A ′ b ′
Page 42 and 43: ⎛( ) ⎛ ⎞⎛⎞⎞⎜ ⎜Σ e
Page 44 and 45: Figure 1: Expected profit for asymm
Page 46 and 47: Figure 3: Trading intensities for a
Page 48 and 49: Figure 5: µ coefficients for asymm
Page 50 and 51: Figure 7: Trading intensity for asy
Page 52: VI ReferencesAdmati, Anat, 1985, A

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