Cross-Asset Speculation in Stock Markets∗ - Econometrics at Illinois ...

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