Financial Market Equilibrium

1.1  Derivation of Lemma 1

To derive the financial-market equilibrium under partly informative prices, match the coefficients in equation (10) with the coefficients in equation (5) and solve the non-linear equation system in the three unknowns (Subscript[π, 0]),(Subscript[π, S]), and (Subscript[π, X]). To make the math more intuitive, I introduce the variable u in addition, as in Appendix D. In the following equations, I  write u, p0, ps, and px instead of (Subscript[π, 0]),(Subscript[π, S]), and (Subscript[π, X]) to keep Mathematica from confusing the parameters with the number π. For similar reasons, the number of signals is defined as nSig, the number of investors as iInv, and the sum of all signal realizations as sumSig.

Solve[{u == 
            (1/(τ^2)) + (1 - λ) ((((λ nSig)/(u (σ^2))) (((λ nSig)/(u (σ^2))) - 1))/(((((λ/(u (σ^2)))))^2) nSig (σ^2) + (((((γ ω)/iInv)))^2) (1/(u^2)))) + λ (nSig/(σ^2))}, {u}];
u = FullSimplify[u /. Extract[Take[%, 1], 1]]

ps = 
      FullSimplify[(1/u) (λ/(σ^2))]
px = FullSimplify[(1/u) (γ/iInv)]
p0 = 
      FullSimplify[(((μ/(τ^2)) + (1 - λ) ((ps nSig  px xbar)/((ps^2) nSig (σ^2) + (px^2) (ω^2))))/(u + (1 - λ) ((ps nSig)/((ps^2) nSig (σ^2) + (px^2) (ω^2)))))]

The covariance between RP and the fundamental, and the covariance response to nSig and λ, is

FullSimplify[((λ (τ^2))/((σ^2) (1 + 
                  nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) 
        nSig (τ^2)]
FullSimplify[
      D[((λ (τ^2))/((σ^2) (1 + 
                    nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) nSig (τ^2), 
        nSig]]
FullSimplify[
      D[((λ (τ^2))/((σ^2) (1 + 
                    nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) 
          nSig (τ^2), λ ]]
FullSimplify[
      D[((λ (τ^2))/((σ^2) (1 + 
                      nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) nSig (τ^2), nSig] - 
        D[((λ (τ^2))/((σ^2) (1 + 
                      nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) 
            nSig (τ^2), λ ]]

The covariance between RP and the sum of the signals, and its response to nSig and λ, is

FullSimplify[((λ (τ^2))/((σ^2) (1 + 
                  nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) 
        nSig ((σ^2) + 
            nSig λ (τ^2))]
FullSimplify[
      D[((λ (τ^2))/((σ^2) (1 + 
                    nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) 
          nSig ((σ^2) + nSig λ (τ^2)), 
        nSig]]
FullSimplify[
      D[((λ (τ^2))/((σ^2) (1 + 
                    nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) 
          nSig ((σ^2) + 
              nSig λ (τ^2)), λ ]]
FullSimplify[
      D[((λ (τ^2))/((σ^2) (1 + 
                      nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) 
            nSig ((σ^2) + nSig λ (τ^2)), nSig] - 
        D[((λ (τ^2))/((σ^2) (1 + 
                      nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) 
            nSig ((σ^2) + 
                nSig λ (τ^2)), λ ]]

The variance of RP, and its response to nSig and λ, is

FullSimplify[(((((λ (τ^2))/((σ^2) (1 + nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2)))))))))^2) 
          nSig ((σ^2) + 
              nSig λ (τ^2)) + ((γ (τ^2))/(iInv (1 + 
                    nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) (ω^2)]
FullSimplify[
      D[(((((λ (τ^2))/((σ^2) (1 + nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2)))))))))^2) 
            nSig ((σ^2) + 
                nSig λ (τ^2)) + ((γ (τ^2))/(iInv (1 + 
                      nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) (ω^2), 
        nSig]]
FullSimplify[
      D[(((((λ (τ^2))/((σ^2) (1 + nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2)))))))))^2) 
            nSig ((σ^2) + 
                nSig λ (τ^2)) + ((γ (τ^2))/(iInv (1 + 
                      nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) (ω^2), λ ]]

The variance of RP, conditional on sumSig, and  its response to nSig and λ is

FullSimplify[((γ (τ^2))/(iInv (1 + 
                  nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) (ω^2)]
FullSimplify[
      D[((γ (τ^2))/(iInv (1 + 
                    nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) (ω^2), 
        nSig]]
FullSimplify[
      D[((γ (τ^2))/(iInv (1 + 
                    nSig λ (((τ^2)/(σ^2)) + (((iInv^2) (-1 + λ))/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))))))) (ω^2), λ ]]

1.2  Solutions for Normalized Excess Return (Appendix E): Decomposition into Updated Variance and the Key Term

The financial-market equilibrium implies closed-form solutions for the pre-posterior expectation and variance of the risky-asset. Pre-posterior beliefs differ for Price Watchers and News Subscribers so that the expectation and variance imply difference normalized excess returns. To facilitate the derivations, I separately consider ((((τ^i)))^2)and ((μ^i)-RP)/((((τ^i)))^2). In this notebook file, I refer to ((((τ^i)))^2) as the Updated Variance and to ((μ^i)-RP)/((((τ^i)))^2) as the Key Term.

newsSubscribersKeyTerm = 
      FullSimplify[(μ/(τ^2)) + 
          (1/(σ^2)) sumSig - ((1/(τ^2)) + 
                (nSig/(σ^2))) (p0  +  ps sumSig  -  
                px X)]

priceWatchersKeyTerm = 
      Simplify[(μ/(τ^2)) - ((ps nSig)/((ps^2) 
                    nSig (σ^2) + (px^2) (ω^2))) (p0  -  
                px xbar) - ((1/(τ^2)) + (((ps^2) (nSig^2) - ps nSig)/((ps^2) nSig (σ^2)  +  (px^2) (ω^2)))) (p0  +  ps sumSig  -  px X)]

With the results for the KeyTerms at hand, check for market clearing in equilibrium to make sure that there was no mistake so far:

FullSimplify[(1 - λ) 
          priceWatchersKeyTerm + λ newsSubscribersKeyTerm]

The updated variances ((((τ^NS)))^2) and ((((τ^PW)))^2) are

tausqNewsSubscribers = 
      FullSimplify[(1/((1/(τ^2)) + (nSig/(σ^2))))]

tausqPriceWatchers = 
      FullSimplify[(((τ^2) ((ps^2) nSig (σ^2) + (px^2) (ω^2)))/((ps^2) nSig ((σ^2) + nSig (τ^2)) + (px^2) (ω^2)))]

1.3  Moments of the Key Term (Appendix F)

expNewsSubscribers = 
      FullSimplify[((γ ((iInv^2) nSig (xbar + xbar (-1 + λ)) λ ((σ^2) + nSig (τ^2)) + γ (σ^2) (-iInv (-1 + λ) (nSig μ - nSig μ) + xbar γ ((σ^2) + nSig (τ^2))) (ω^2)))/((iInv^3) nSig (λ^2) ((σ^2) + nSig (τ^2)) + iInv (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2)))]
varNewsSubscribers = 
      FullSimplify[(((((γ ((iInv^2) nSig λ ((σ^2) + nSig (τ^2))) + (γ^2) (σ^2) (γ ((σ^2) + nSig (τ^2))) (ω^2)))^2) (ω^2) + (γ^4) (σ^4) (((-iInv (-1 + λ)))^2) (ω^4) nSig ((σ^2) + nSig (τ^2)))/((((iInv^3) nSig (λ^2) ((σ^2) + nSig (τ^2)) + iInv (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2)))^2))]

expPriceWatchers = 
      FullSimplify[((γ ((iInv^2) nSig xbar (λ^2) ((σ^2) + nSig (τ^2)) + iInv γ λ (-nSig μ + nSig μ) (σ^2) (ω^2) + xbar (γ^2) (σ^4) (ω^2)))/(iInv ((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2))))]
varPriceWatchers = 
      FullSimplify[((((( (γ^3) (σ^4) (ω^2)))^2) (ω^2) + ((( iInv (γ^2) λ  (σ^2) (ω^2)))^2) nSig ((σ^2) + nSig (τ^2)))/(((iInv ((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2))))^2))]