Properties of Information Market Equilibrium under Partly Informative Price

3.1  Derivation of Terms in Table 2 (Utility Responses to Signal Acquisition, Appendix H)

To derive the terms in Table 2, combine results from the previous sections 2.1 and 2.2 in the preceding notebook infgauss2.nb. The results were

Combine the results to find

focEtermNewsSubscribers = 
      FullSimplify[(1/nSig) ((tausqNewsSubscribers (expNewsSubscribers^2))/(1 + (tausqNewsSubscribers/(1 + R)) 
                  varNewsSubscribers)) ((1/2) 
                elasttauNewsSubscribers + 
              elastexpNewsSubscribers)]
focVartermNewsSubscribers = 
      FullSimplify[(1/nSig) (((tausqNewsSubscribers/(1 + R)) 
                varNewsSubscribers)/(((1 + (tausqNewsSubscribers/(1 + R)) varNewsSubscribers))^2)) ((1/2) 
                elasttauNewsSubscribers + (1/2) 
                elastvarNewsSubscribers)]
focDelfactorNewsSubscribers = 
      FullSimplify[(1 + R) + tausqNewsSubscribers varNewsSubscribers - 
          tausqNewsSubscribers (expNewsSubscribers^2)]

focEtermPriceWatchers = 
      FullSimplify[(1/nSig) ((tausqPriceWatchers (expPriceWatchers^2))/(1 + (tausqPriceWatchers/(1 + R)) 
                  varPriceWatchers)) ((1/2) elasttauPriceWatchers + 
              elastexpPriceWatchers)]
focVartermPriceWatchers = 
      FullSimplify[(1/nSig) ((tausqPriceWatchers varPriceWatchers)/(((1 + (tausqPriceWatchers/(1 + R)) varPriceWatchers))^2)) ((1/2) elasttauPriceWatchers + (1/2) 
                elastvarPriceWatchers)]
focDelfactorPriceWatchers = 
      FullSimplify[(1 + R) + tausqPriceWatchers varPriceWatchers - 
          tausqPriceWatchers (expPriceWatchers^2)]

To compare the terms in Table 2 between News Subscribers and Price Watchers, combine them:

FullSimplify[
      focEtermNewsSubscribers - focEtermPriceWatchers]
  
FullSimplify[
      focDelfactorNewsSubscribers - focDelfactorPriceWatchers]

The cutoff market size for a positive focDelfactorNewsSubscribers is

FullSimplify[
      Solve[R + ((((σ^2) + nSig (τ^2)) ((iInv^2) nSig (λ^2) + (γ^2) (σ^2) (ω^2)) ((iInv^4) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^4) (σ^4) (τ^2) (ω^2) (-xbarsq + (ω^2)) + (iInv^2) (γ^2) (σ^2) ((σ^2) (ω^2) + nSig λ (τ^2) (-xbarsq λ + 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)))^2)) == 0, {xbarsq}]]

At this cutoff  market size for a positive focDelfactorNewsSubscribers, the focEtermNewsSubscribers is

FullSimplify[focEtermNewsSubscribers /. %]

The  cutoff market size for a positive focDelfactorPriceWatchers is

FullSimplify[
      Solve[R + ((((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^4) (ω^2)) ((iInv^4) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^4) (σ^4) (τ^2) (ω^2) (-xbarsq + (ω^2)) + (iInv^2) (γ^2) (σ^2) ((σ^2) (ω^2) + nSig λ (τ^2) (-xbarsq λ + 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)))^2)) == 0, {xbarsq}]]

The difference between the two cutoff market sizes for a positive focDelfactorNewsSubscribers and focDelfactorPriceWatchers is

FullSimplify[(1/((γ^2) (σ^2) (τ^2) ((σ^2) + nSig (τ^2)) ((((iInv^2) nSig (λ^2) + (γ^2) (σ^2) (ω^2)))^2))) ((iInv^6) (nSig^2) (1 + 
                    R) (λ^4) ((((σ^2) + nSig (τ^2)))^2) + 
              (iInv^4) nSig (γ^2) (λ^2) (σ^2) ((σ^2) + nSig (τ^2)) (2 (1 + R) (σ^2) + 
                    nSig (1 + 
                          2 (1 + 
                                R) λ) (τ^2)) (ω^2) + (iInv^2) (γ^4) (σ^4) ((1 + R) (σ^4) + 
                    nSig (1 + λ (2 + 
                                2 R + λ)) (σ^2) (τ^2) + (nSig^2) λ (2 + λ + 
                          R λ) (τ^4)) (ω^4) + (γ^6) (σ^6) (τ^2) ((σ^2) + 
                    nSig (τ^2)) (ω^6)) - (((iInv^6) (nSig^2) (1 + R) (λ^4) ((((σ^2) + nSig (τ^2)))^2) + 2 (iInv^4) nSig (1 + R) (γ^2) (λ^2) (σ^2) ((σ^2) + nSig (τ^2)) ((σ^2) + nSig λ (τ^2)) (ω^2) + (iInv^2) (γ^4) (σ^4) ((1 + R) (σ^4) + nSig λ (2 + 2 R + λ) (σ^2) (τ^2) + (nSig^2) (1 + R) (λ^2) (τ^4)) (ω^4) + (γ^6) (σ^8) (τ^2) (ω^6))/((γ^2) (σ^2) (τ^2) ((iInv^2) nSig (λ^2) + (γ^2) (σ^2) (ω^2)) ((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^4) (ω^2))))]

The cutoff market size for a positive difference  focDelfactorNewsSubscribers - focDelfactorPriceWatchers is

FullSimplify[
      Solve[((nSig (γ^2) (σ^2) (τ^2) (ω^2) ((iInv^4) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^4) (σ^4) (τ^2) (ω^2) (-xbarsq + (ω^2)) + (iInv^2) (γ^2) (σ^2) ((σ^2) (ω^2) + nSig λ (τ^2) (-xbarsq λ + 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)))^2)) == 0, {xbarsq}]]

2.5  Limits in Table 3

The limits for Table 3 are

FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers, ω -> 0]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers, ω -> ∞]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers, γ -> 0]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers, γ -> ∞]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers, σ -> 0]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers, σ -> ∞]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + elastexpNewsSubscribers, 
        xbar -> 0]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + elastexpNewsSubscribers, 
        xbar -> ∞]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers, τ -> 0]] 
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers, τ -> ∞]]

FullSimplify[
      Solve[-(1/2) + (σ^2) (-(1/(2 ((σ^2) + 
                              nSig (τ^2)))) + 
                  (1/((σ^2) + nSig λ (τ^2)))) == 
          0, {λ}]]
FullSimplify[
      Solve[-(1/2) + (σ^2) (-(1/(2 ((σ^2) + 
                              nSig (τ^2)))) + 
                  (1/((σ^2) + nSig λ (τ^2)))) == 
          0, {nSig}]]
FullSimplify[
      Solve[-(1/2) + 
            (iInv^2) nSig λ (-(1/((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2))) + (λ/((iInv^2) nSig (λ^2) + (γ^2) (σ^2) (ω^2)))) == 
          0, {nSig}]]
FullSimplify[
      Solve[(-3 + (Sqrt[(-9 + λ) (-1 + λ)]) + λ)  == 
          0, {λ}]]
(-3 + (Sqrt[(-9 + λ) (-1 + λ)]) + λ)  /. λ ->  .5

FullSimplify[
      Limit[focEtermNewsSubscribers, ω -> 0]]
FullSimplify[
      Limit[focEtermNewsSubscribers, ω -> ∞]]
FullSimplify[
      Limit[focEtermNewsSubscribers, γ -> 0]]
FullSimplify[
      Limit[focEtermNewsSubscribers, γ -> ∞]]
FullSimplify[
      Limit[focEtermNewsSubscribers, σ -> 0]]
FullSimplify[
      Limit[focEtermNewsSubscribers, σ -> ∞]]
FullSimplify[Limit[focEtermNewsSubscribers, xbar -> 0]]
FullSimplify[
      Limit[focEtermNewsSubscribers, 
        xbar -> ∞]]
FullSimplify[
      Limit[focEtermNewsSubscribers, τ -> 0]]
FullSimplify[
      Limit[focEtermNewsSubscribers, τ -> ∞]]

FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, ω -> 0]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, ω -> ∞]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, γ -> 0]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, γ -> ∞]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, σ -> 0]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, σ -> ∞]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, xbar -> 0]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + elastexpNewsSubscribers, 
        xbar -> ∞]]
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, τ -> 0]] 
FullSimplify[
      Limit[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers, τ -> ∞]]

FullSimplify[
      Limit[focVartermNewsSubscribers focDelfactorPriceWatchers, ω -> 0]]
FullSimplify[
      Limit[focVartermNewsSubscribers focDelfactorPriceWatchers, ω -> ∞]]
FullSimplify[
      Limit[focVartermNewsSubscribers focDelfactorPriceWatchers, γ -> 0]]
FullSimplify[
      Limit[focVartermNewsSubscribers focDelfactorPriceWatchers, γ -> ∞]]
FullSimplify[
      Limit[focVartermNewsSubscribers focDelfactorPriceWatchers, σ -> 0]]
FullSimplify[
      Limit[focVartermNewsSubscribers focDelfactorPriceWatchers, σ -> ∞]]
\(FullSimplify[
      Limit[focVartermNewsSubscribers\ focDelfactorPriceWatchers, 
        xbar -> 0]]
 (*\ 
      FullSimplify[
        Limit[focVartermNewsSubscribers\ focDelfactorPriceWatchers, 
          xbar -> ∞]]\ *) \)
FullSimplify[
      Limit[focVartermNewsSubscribers focDelfactorPriceWatchers, τ -> 0]]
FullSimplify[
      Limit[focVartermNewsSubscribers focDelfactorPriceWatchers, τ -> ∞]]

FullSimplify[-(((1 + 
                    R) (xbar^2) (τ^2) ((-1 + 
                          2 λ) (σ^2) + 
                    nSig λ (τ^2)))/(2 ((σ^2) + 
                    nSig (τ^2)) ((σ^2) + 
                    nSig λ (τ^2)) (ω^2))) + (((1 + R) (σ^2) (τ^2) ((-1 + 2 λ) (σ^2) + nSig λ (τ^2)) (xbar - ω) (xbar + ω))/(2 ((((σ^2) + nSig (τ^2)))^2) ((σ^2) + nSig λ (τ^2)) (ω^2)))]
FullSimplify[
      Solve[-(((iInv^2) (1 + 
                      R) (γ^2) (λ^2) (σ^2) (ω^2) ((iInv^4) (nSig^2) (1 + 
                            R) (λ^2) + (γ^4) (σ^4) (ω^2) (-xbarsq + (1 + R) (ω^2)) + 
                      (iInv^2) nSig (γ^2) λ (σ^2) (-xbarsq λ + 2 (1 + 
                                  R) (ω^2))))/(2 ((((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2)))^2) ((((iInv^2) nSig (1 + R) (λ^2) + (γ^2) (σ^2) (ω^2)))^2))) == 0, {xbarsq}]]

2.5  Proof of Theorem 1 (Appendix I)

The crucial part for the proof of Theorem 6 is the derivation of the zero-point. Note that both focEtermPricewatchers and focDelfactorPricewatchers depend on \(TraditionalForm\`x\&_\^2\). To make Mathematica solve for \(TraditionalForm\`\(x\&_\)\_0\%2\), it is helpful to substitute the variable xbarsq for \(TraditionalForm\`x\&_\^2\). This is done below so that the value of \(TraditionalForm\`\(x\&_\)\_0\%2\), as reported in appendix G, can be derived as

FullSimplify[
        D[focEtermNewsSubscribers, ω]]/(1 + 
          R)
FullSimplify[
        D[focVartermNewsSubscribers focDelfactorPriceWatchers, ω]]/(1 + R)

2.6  Proof of Theorem 2 (Appendix I)

The crucial part for the proof of Theorem 6 is the derivation of the zero-point. Note that both focEtermPricewatchers and focDelfactorPricewatchers depend on \(TraditionalForm\`x\&_\^2\). To make Mathematica solve for \(TraditionalForm\`\(x\&_\)\_0\%2\), it is helpful to substitute the variable xbarsq for \(TraditionalForm\`x\&_\^2\). This is done below so that the value of \(TraditionalForm\`\(x\&_\)\_0\%2\), as reported in appendix G, can be derived as

focEtermPriceWatchersRewr = -((1 + 
                    R) xbarsq (γ^2) λ (σ^2) (τ^4) ((iInv^6) (nSig^3) (λ^5) ((σ^2) + nSig (τ^2)) + 
                    (iInv^4) (nSig^2) (γ^2) (λ^3) (σ^2) (3 (σ^2) + nSig (2 - λ) (τ^2)) (ω^2) + 
                    2 (iInv^2) nSig (γ^4) λ (1 + λ) (σ^6) (ω^4) + 
                    2 (γ^6) (σ^8) (ω^6)))/(2 ((iInv^2) nSig (λ^2) ((σ^2) + 
                        nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + 
                        nSig λ (τ^2)) (ω^2)) ((iInv^6) (nSig^2) (1 + 
                        R) (λ^4) ((((σ^2) + nSig (τ^2)))^2) + 2 (iInv^4) nSig (1 + 
                        R) (γ^2) (λ^2) (σ^2) ((σ^2) + nSig (τ^2)) ((σ^2) + 
                        nSig λ (τ^2)) (ω^2) + 
                  (iInv^2) (γ^4) (σ^4) ((1 + 
                              R) (σ^4) + 
                        nSig λ (2 + 
                              2 R + λ) (σ^2) (τ^2) + 
                        (nSig^2) (1 + 
                              R) (λ^2) (τ^4)) (ω^4) + (γ^6) (σ^8) (τ^2) (ω^6)));
focDelfactorPriceWatchersRewr = ((R + ((((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^4) (ω^2)) ((iInv^4) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^4) (σ^4) (τ^2) (ω^2) (-xbarsq + (ω^2)) + (iInv^2) (γ^2) (σ^2) ((σ^2) (ω^2) + nSig λ (τ^2) (-xbarsq λ + 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)))^2)))/(1 + (((γ^4) (σ^6) (τ^2) (ω^4) ((iInv^2) nSig (λ^2) + (γ^2) (σ^2) (ω^2)))/((1 + R) ((((iInv^3) nSig (λ^2) ((σ^2) + nSig (τ^2)) + iInv (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2)))^2)))));
FullSimplify[
      Solve[{focEtermPriceWatchersRewr + 
              focDelfactorPriceWatchersRewr focVartermPriceWatchers == 
            0}, {xbarsq}]]
FullSimplify[
      Limit[xbarsq /. Extract[Take[%, 1], 1], ω -> 
          0]]
FullSimplify[
      Limit[xbarsq /. 
          Extract[Take[%%, 1], 1], ω -> ∞]]

2.7  Limits for Proofs of Theorems 3 and 4

Take the limits of the News Watchers' first order condition as needed for theorems 7 and 8 by letting λ approach the extremes.

FullSimplify[
        Limit[focEtermNewsSubscribers + 
            focVartermNewsSubscribers focDelfactorPriceWatchers, λ -> 0]]/(1 + R)
FullSimplify[
        Limit[focEtermNewsSubscribers + 
            focVartermNewsSubscribers focDelfactorPriceWatchers, 
          nSig -> 0]]/(1 + R)
FullSimplify[
        Limit[focEtermNewsSubscribers + 
            focVartermNewsSubscribers focDelfactorPriceWatchers, λ -> 1]]/(1 + R)

(((τ^2) ((iInv^4) (1 + R) (σ^2) + (γ^4) (τ^4) (ω^2) (nSig (xbar^2) (τ^2) + (σ^2) (ω^2)) + (iInv^2) (γ^2) (τ^2) ((xbar^2) (R (σ^2) + nSig (τ^2)) + (2 + R) (σ^2) (ω^2))))/(2 ((((iInv^2) ((1 + R) (σ^2) + nSig (τ^2)) + (γ^2) (τ^2) ((σ^2) + nSig (τ^2)) (ω^2)))^2))) /. 
      nSig -> 0
(( (τ^2) ((iInv^4) (1 + R) (1 - (λ^2)) + (γ^4) (1 - 2 λ) (τ^4) (ω^4) - (iInv^2) (γ^2) (τ^2) ((xbar^2) (2 - λ) λ - (2 - λ (2 + λ)) (ω^2) - R (1 - 2 λ) ((xbar^2) + (ω^2)))))/(2 (σ^2) ((((iInv^2) (1 + R) + (γ^2) (τ^2) (ω^2)))^2))) /. λ -> 
        0