Properties of Information Market Equilibrium under Partly Informative Price

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

To evaluate the terms in Table 2, revisit results from the preceding notebook infgauss2.nb. The results were

expNewsSubscribers = ((xbar γ ((σ^2) + nSig (τ^2)) ((iInv^2) nSig (λ^2) + (γ^2) (σ^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 = (((γ^2) ((σ^2) + nSig (τ^2)) (ω^2) ((iInv^2) nSig (γ^2) (((-1 + λ))^2) (σ^4) (ω^2) + ((σ^2) + nSig (τ^2)) ((((iInv^2) nSig λ + (γ^2) (σ^2) (ω^2)))^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));
tausqNewsSubscribers = 
        (1/((nSig/(σ^2)) + (1/(τ^2))));

elastexpNewsSubscribers = ((nSig (γ^2) (-1 + λ) (σ^2) (τ^2) (ω^2) ((iInv^2) (nSig^2) (λ^2) (τ^2) - (γ^2) (σ^4) (ω^2)))/(((σ^2) + nSig (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^2) ((σ^2) + nSig λ (τ^2)) (ω^2))));
elastvarNewsSubscribers = -((nSig (γ^2) (-1 + λ) (σ^4) (ω^2) ((iInv^4) nSig (λ^2) (1 + λ) ((σ^2) + nSig (τ^2)) ((σ^2) + 
                          2 nSig (τ^2)) + 
                    (iInv^2) (γ^2) (σ^2) ((1 + λ) (σ^4) + 
                          nSig (2 + λ (5 + λ)) (σ^2) (τ^2) + 6 (nSig^2) λ (τ^4)) (ω^2) + 
                    2 (γ^4) (σ^4) (τ^2) ((σ^2) + 
                          nSig (τ^2)) (ω^4)))/(((σ^2) + nSig (τ^2)) ((iInv^2) nSig (λ^2) ((σ^2) + 
                          nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + 
                          nSig λ (τ^2)) (ω^2)) ((iInv^4) (nSig^2) (λ^2) ((σ^2) + nSig (τ^2)) + 
                    (iInv^2) nSig (γ^2) (σ^2) ((1 + (λ^2)) (σ^2) + 
                          2 nSig λ (τ^2)) (ω^2) + (γ^4) (σ^4) ((σ^2) + 
                          nSig (τ^2)) (ω^4))));
elasttauNewsSubscribers = -((nSig (τ^2))/((σ^2) + 
                nSig (τ^2)));
focEtermNewsSubscribers = -((1 + 
                    R) (xbar^2) (γ^2) (σ^2) (τ^4) ((iInv^2) nSig (λ^2) + (γ^2) (σ^2) (ω^2)) ((iInv^4) (nSig^2) (λ^4) ((σ^2) + nSig (τ^2)) + 
                    (iInv^2) nSig (γ^2) (λ^2) (σ^2) (2 (σ^2) - 
                          nSig (-3 + λ) (τ^2)) (ω^2) + (γ^4) (σ^4) ((-1 + 
                                2 λ) (σ^2) + 
                          nSig λ (τ^2)) (ω^4)))/(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) + (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)));
focVartermNewsSubscribers = -((iInv^2) (1 + 
                    R) (γ^2) (σ^2) (τ^2) (ω^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^3) (λ^4) (τ^2) ((σ^2) + 
                          nSig (τ^2)) + 
                    (iInv^4) nSig (γ^2) (λ^2) (σ^2) ((-1 + (λ^2)) (σ^4) + 
                          3 nSig (λ^2) (σ^2) (τ^2) + 
                          3 (nSig^2) λ (τ^4)) (ω^2) + 
                    (iInv^2) (γ^4) (σ^4) ((-1 + (λ^2)) (σ^4) + 
                          nSig λ (-3 + λ (5 + λ)) (σ^2) (τ^2) + 
                          3 (nSig^2) (λ^2) (τ^4)) (ω^4) + (γ^6) (σ^6) (τ^2) ((-1 + 
                                2 λ) (σ^2) + 
                          nSig λ (τ^2)) (ω^6)))/(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)))^2));
focDelfactorNewsSubscribers = 
        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) (-(xbar^2) + (ω^2)) + (iInv^2) (γ^2) (σ^2) ((σ^2) (ω^2) + nSig λ (τ^2) (-(xbar^2) 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));

expPriceWatchers = (((iInv^2) nSig xbar γ (λ^2) ((σ^2) + nSig (τ^2)) + xbar (γ^3) (σ^4) (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)));
varPriceWatchers = (((iInv^2) nSig (γ^4) (λ^2) (σ^4) ((σ^2) + nSig (τ^2)) (ω^4) + (γ^6) (σ^8) (ω^6))/((((iInv^3) nSig (λ^2) ((σ^2) + nSig (τ^2)) + iInv (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2)))^2));
tausqPriceWatchers = (((τ^2) ((iInv^2) nSig (λ^2) (σ^2) + (γ^2) (σ^4) (ω^2)))/((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^4) (ω^2)));

elastexpPriceWatchers = (((iInv^2) (nSig^3) (γ^2) (λ^3) (σ^2) (τ^4) (ω^2) - nSig (γ^4) λ (σ^6) (τ^2) (ω^4))/(((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^4) (ω^2)) ((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2))));
elastvarPriceWatchers = ((nSig λ (-(iInv^4) nSig (λ^3) ((σ^2) + nSig (τ^2)) ((σ^2) + 2 nSig (τ^2)) - (iInv^2) (γ^2) λ (σ^4) ((σ^2) + nSig (2 + λ) (τ^2)) (ω^2) - 2 (γ^4) (σ^6) (τ^2) (ω^4)))/(((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^4) (ω^2)) ((iInv^2) nSig (λ^2) ((σ^2) + nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2))));
elasttauPriceWatchers = -(((nSig^2) (τ^2) ((iInv^4) nSig (λ^4) + 
                    2 (iInv^2) (γ^2) (λ^2) (σ^2) (ω^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))));
focEtermPriceWatchers = -((1 + 
                    R) (xbar^2) (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)));
focVartermPriceWatchers = -(((iInv^2) (((1 + R))^2) (γ^4) λ (σ^6) (τ^2) (ω^4) ((iInv^2) nSig (λ^2) ((σ^2) + 
                          nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + 
                          nSig λ (τ^2)) (ω^2)) ((iInv^4) nSig (λ^3) ((σ^2) + 3 nSig (τ^2)) + 
                    (iInv^2) (γ^2) λ (σ^2) ((σ^2) + nSig (4 + λ) (τ^2)) (ω^2) + 
                    2 (γ^4) (σ^4) (τ^2) (ω^4)))/(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)))^2)));
focDelfactorPriceWatchers = 
        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) (-(xbar^2) + (ω^2)) + (iInv^2) (γ^2) (σ^2) ((σ^2) (ω^2) + nSig λ (τ^2) (-(xbar^2) 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));

The first derivative of focEtermNewsSubscribers with respect to is (xbar^2) (xbarsq) is

FullSimplify[
      D[-((1 + 
                    R) xbarsq (γ^2) (σ^2) (τ^4) ((iInv^2) nSig (λ^2) + (γ^2) (σ^2) (ω^2)) ((iInv^4) (nSig^2) (λ^4) ((σ^2) + nSig (τ^2)) + 
                    (iInv^2) nSig (γ^2) (λ^2) (σ^2) (2 (σ^2) - 
                          nSig (-3 + λ) (τ^2)) (ω^2) + (γ^4) (σ^4) ((-1 + 
                                2 λ) (σ^2) + 
                          nSig λ (τ^2)) (ω^4)))/(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) + (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))), 
        xbarsq]]

The restriction on  (ω^2) (omegasq) for a positive derivative is

FullSimplify[
      Solve[((iInv^4) (nSig^2) (λ^4) ((σ^2) + 
                    nSig (τ^2)) + 
              (iInv^2) nSig (γ^2) (λ^2) (σ^2) (2 (σ^2) - nSig (-3 + λ) (τ^2)) 
                omegasq + (γ^4) (σ^4) ((-1 + 
                          2 λ) (σ^2) + 
                    nSig λ (τ^2)) (omegasq^2)) == 0, { 
          omegasq}]]

where the former root is negative  so that only the latter root is viable.

FullSimplify[
      Solve[(-2 (σ^2) - 
              nSig (3 - λ) (τ^2) - (Sqrt[1 - λ]) (Sqrt[8 (σ^4) + 16 nSig (σ^2) (τ^2) + (nSig^2) (9 - λ) (τ^4)])) == 0, {λ}]]
FullSimplify[
      Solve[(-2 (σ^2) - 
              nSig (3 - λ) (τ^2) - (Sqrt[1 - λ]) (Sqrt[8 (σ^4) + 16 nSig (σ^2) (τ^2) + (nSig^2) (9 - λ) (τ^4)])) == 0, {nSig}]]
FullSimplify[
      D[(-2 (σ^2) - 
              nSig (3 - λ) (τ^2) - (Sqrt[1 - λ]) (Sqrt[8 (σ^4) + 16 nSig (σ^2) (τ^2) + (nSig^2) (9 - λ) (τ^4)])), nSig] /. 
        nSig -> (((1 - 2 λ) (σ^2))/(λ (τ^2)))]

At \(nSig -> \(\((1 - 2\ λ)\)\ σ\^2\)\/\(λ\ τ\^2\)\), focVartermNewsSubscribers evaluates to

FullSimplify[
      focVartermNewsSubscribers /. 
        nSig -> (((1 - 2 λ) (σ^2))/(λ (τ^2)))]

The cutoff supply noise  (ω^2) (omegasq) for focVartermNewsSubscribers to be positive is

FullSimplify[Solve[ - 
                  (iInv^6) (nSig^3) (λ^4) (τ^2) ((σ^2) + nSig (τ^2))+(iInv^4) nSig (γ^2) (λ^2) (σ^2) ((1 - (λ^2)) (σ^4) - 
                    3 nSig (λ^2) (σ^2) (τ^2) - 
                    3 (nSig^2) λ (τ^4)) omegasq+(iInv^2) (γ^4) (σ^4) ((1 - (λ^2)) (σ^4) + 
                    nSig λ (3 - λ (5 + λ)) (σ^2) (τ^2) - 3 (nSig^2) (λ^2) (τ^4)) (omegasq^2)-(γ^6) (σ^6) (τ^2) ((1 - 2 λ) (σ^2) - 
                    nSig λ (τ^2)) (omegasq^3)==0,{omegasq}]]

2.6  Proof of Theorem 1 (Appendix I)

To assess the utility effect of signals on Price Watchers, it suffices to evaluate the term d + b (d - g), where  b = (tausqPriceWatchers/(1 + R)) varPriceWatchers,  d = (1/nSig) ((1/2) elasttauPriceWatchers + 
              elastexpPriceWatchers),  and g = (1/nSig) ((1/2) elasttauPriceWatchers + (1/2) 
                elastvarPriceWatchers).

FullSimplify[(1/nSig) ((1/2) elasttauPriceWatchers + 
              elastexpPriceWatchers) + (1/nSig) (tausqPriceWatchers/(1 + R)) 
          varPriceWatchers (((1/2) elasttauPriceWatchers + 
                  elastexpPriceWatchers) - ((1/2) 
                    elasttauPriceWatchers + (1/2) 
                    elastvarPriceWatchers))]

So the term d + b (d - g) is strictly negative, so a signal to News Subscribers inflicts a strict negative externality on Price Watchers. A direct evaluation of Price Watchers' marginal utility is possible.

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));
FullSimplify[
      Solve[{focEtermPriceWatchersRewr + 
              focDelfactorPriceWatchersRewr focVartermPriceWatchers == 
            0}, {xbarsq}]]
FullSimplify[
      Limit[xbarsq /. Extract[Take[%, 1], 1], ω -> 
          0]]
FullSimplify[
      Limit[xbarsq /. 
          Extract[Take[%%, 1], 1], ω -> ∞]]

2.5  Proof of Theorem 2 (Appendix J)

Evaluate the term D + B (D - G), where  B = (tausqNewsSubscribers/(1 + R)) varNewsSubscribers,  D = (1/nSig) ((1/2) elasttauNewsSubscribers + 
              elastexpNewsSubscribers),  and G = (1/nSig) ((1/2) elasttauNewsSubscribers + (1/2) 
                elastvarNewsSubscribers).

criterionPosUtil = 
      FullSimplify[(1/nSig) ((1/2) elasttauNewsSubscribers + 
                elastexpNewsSubscribers) + (tausqNewsSubscribers/(1 + 
                  R)) varNewsSubscribers ((1/nSig) ((1/2) 
                        elasttauNewsSubscribers + 
                      elastexpNewsSubscribers) - (1/nSig) ((1/2) 
                        elasttauNewsSubscribers + (1/2) 
                        elastvarNewsSubscribers))]

FullSimplify[
      Limit[criterionPosUtil, ω -> 0]]
FullSimplify[
      Limit[criterionPosUtil, ω -> ∞]]

The above term is strictly positive iff (rewriting the signs in  D + B (D - G) for clarity)

condThm2versionA=((iInv^6) (nSig^3) (1 + 
                  R) (λ^6) ((((σ^2) + nSig (τ^2)))^2) + 3 (iInv^4) (nSig^2) (1 + 
                  R) (γ^2) (λ^4) (σ^2) ((((σ^2) + nSig (τ^2)))^2) (ω^2) + 
            (iInv^2) nSig (γ^4) (λ^2) (σ^4) ((2 + 
                        R + 2 (1 + 
                              R) λ - (λ^2)) (σ^4) + 2 nSig (2 + R + 2 (1 + 
                              R) λ - (λ^2)) (σ^2) (τ^2) + (nSig^2) (2 + λ (2 + 
                              R (4 - λ) - λ)) (τ^4)) (ω^4) + (γ^6) (σ^6) ((2 - λ) λ ((((σ^2) + nSig (τ^2)))^2) - 
                  R ((σ^2) + 
                        nSig λ (τ^2)) ((1 - 
                              2 λ) (σ^2) - 
                        nSig λ (τ^2))) (ω^6));

is strictly positive. The last condition strictly increases in (ω^2) (omegasq) if

FullSimplify[D[((iInv^6) (nSig^3) (1 + 
                    R) (λ^6) ((((σ^2) + nSig (τ^2)))^2)+3 (iInv^4) (nSig^2) (1 + R) (γ^2) (λ^4) (σ^2) ((((σ^2) + nSig (τ^2)))^2) omegasq+(iInv^2) nSig (γ^4) (λ^2) (σ^4) ((2 + R + 
                          2 (1 + 
                                R) λ - (λ^2)) (σ^4) + 2 nSig (2 + R + 
                          2 (1 + 
                                R) λ - (λ^2)) (σ^2) (τ^2) + 
                    (nSig^2) (2 + λ (2 + 
                                R (4 - λ) - λ)) (τ^4)) (omegasq^2)+(γ^6) (σ^6) ((2 - λ) λ ((((σ^2) + nSig (τ^2)))^2) - 
                    R ((σ^2) + 
                          nSig λ (τ^2)) ((1 - 
                                2 λ) (σ^2) - 
                          nSig λ (τ^2))) (omegasq^3)),omegasq]]

is strictly positive. Two necessary conditions for the above term to be strictly positive are that the nSig cutoff and the cutoff value for  (ω^2) (omegasq) satisfy

Solve[(1 - 2 λ) (σ^2) - nSig λ (τ^2) == 
        0, {nSig}]
FullSimplify[
      Solve[(iInv^6) (nSig^3) (1 + 
                  R) (λ^6) ((((σ^2) + nSig (τ^2)))^2) + 3 (iInv^4) (nSig^2) (1 + 
                  R) (γ^2) (λ^4) (σ^2) ((((σ^2) + nSig (τ^2)))^2) omegasq + 
            (iInv^2) nSig (γ^4) (λ^2) (σ^4) ((2 + 
                        R + 2 (1 + 
                              R) λ - (λ^2)) (σ^4) + 2 nSig (2 + R + 2 (1 + 
                              R) λ - (λ^2)) (σ^2) (τ^2) - (nSig^2) (-2 + λ (-2 + 
                              R (-4 + λ) + λ)) (τ^4)) (omegasq^2) + (γ^6) (σ^6) (-(-2 + λ) λ ((((σ^2) + nSig (τ^2)))^2) + 
                  R ((σ^2) + 
                        nSig λ (τ^2)) ((-1 + 
                              2 λ) (σ^2) + 
                        nSig λ (τ^2))) (omegasq^3) == 
          0, {omegasq}]]

Note that  D < G for any interior parameter value:

FullSimplify[(1/nSig) ((1/2) elasttauNewsSubscribers + 
              elastexpNewsSubscribers) - (1/nSig) ((1/2) 
                elasttauNewsSubscribers + (1/2) 
                elastvarNewsSubscribers)]

Note that G  >  0 iff

FullSimplify[(1/nSig) ((1/2) elasttauNewsSubscribers + (1/2) 
              elastvarNewsSubscribers)]

is strictly positive. So, the condition G  >  0 is satisfied iff

condThm2versionB=FullSimplify[-((τ^2) ((iInv^2) nSig (λ^2) ((σ^2) + 
                          nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + 
                          nSig λ (τ^2)) (ω^2)) ((iInv^4) (nSig^2) (λ^2) ((σ^2) + nSig (τ^2)) + 
                    (iInv^2) nSig (γ^2) (σ^2) ((1 + (λ^2)) (σ^2) + 
                          2 nSig λ (τ^2)) (ω^2) + (γ^4) (σ^4) ((σ^2) + 
                          nSig (τ^2)) (ω^4)) + (γ^2) (-1 + λ) (σ^4) (ω^2) ((iInv^4) nSig (λ^2) (1 + λ) ((σ^2) + 
                          nSig (τ^2)) ((σ^2) + 
                          2 nSig (τ^2)) + 
                    (iInv^2) (γ^2) (σ^2) ((1 + λ) (σ^4) + 
                          nSig (2 + λ (5 + λ)) (σ^2) (τ^2) + 6 (nSig^2) λ (τ^4)) (ω^2) + 
                    2 (γ^4) (σ^4) (τ^2) ((σ^2) + 
                          nSig (τ^2)) (ω^4)))]

is strictly positive.

Note that D  > 0 iff

FullSimplify[(1/nSig) ((1/2) elasttauNewsSubscribers + 
            elastexpNewsSubscribers)]

is strictly positive. So, the condition D  > 0 is satisfied iff

condThm2versionB=FullSimplify[-((iInv^2) nSig (λ^2) + (γ^2) (σ^2) (ω^2)) ((iInv^2) nSig (λ^2) ((σ^2) + 
                      nSig (τ^2)) + (γ^2) (σ^2) ((σ^2) + nSig λ (τ^2)) (ω^2)) + 
          2 (γ^2) (-1 + λ) (σ^2) (ω^2) ((iInv^2) (nSig^2) (λ^2) (τ^2) - (γ^2) (σ^4) (ω^2))]

is strictly positive.

Check for equality of condThm2versionA and condThm2versionB:

FullSimplify[condThm2versionA - condThm2versionB]

Note that the (ω^2) (omegasq) cutoff value for G  >  0 to be satisfied is

FullSimplify[Solve[((σ^2) + nSig (τ^2)) ((iInv^6) (nSig^3) (λ^4) (τ^2) ((σ^2) + nSig (τ^2))+(iInv^4) nSig (γ^2) (λ^2) (σ^2) ((-1 + (λ^2)) (σ^4) + 
                        3 nSig (λ^2) (σ^2) (τ^2) + 
                        3 (nSig^2) λ (τ^4)) omegasq+(iInv^2) (γ^4) (σ^4) ((-1 + (λ^2)) (σ^4) + 
                        nSig λ (-3 + λ (5 + λ)) (σ^2) (τ^2) + 
                        3 (nSig^2) (λ^2) (τ^4))(omegasq^2)+(γ^6) (σ^6) (τ^2) ((-1 + 2 λ) (σ^2) + 
                        nSig λ (τ^2)) (omegasq^3))==0,{omegasq}]]

Evaluate the expression  D + B (D - G)at the cutoff for  G  >  0:

FullSimplify[((iInv^6) (nSig^3) (1 + 
                  R) (λ^6) ((((σ^2) + nSig (τ^2)))^2) + 3 (iInv^4) (nSig^2) (1 + 
                  R) (γ^2) (λ^4) (σ^2) ((((σ^2) + nSig (τ^2)))^2) omegasq + 
            (iInv^2) nSig (γ^4) (λ^2) (σ^4) ((2 + 
                        R + 2 (1 + 
                              R) λ - (λ^2)) (σ^4) + 2 nSig (2 + R + 2 (1 + 
                              R) λ - (λ^2)) (σ^2) (τ^2) + (nSig^2) (2 + λ (2 + 
                              R (4 - λ) - λ)) (τ^4)) (omegasq^2) + (γ^6) (σ^6) ((2 - λ) λ ((((σ^2) + nSig (τ^2)))^2) - 
                  R ((σ^2) + 
                        nSig λ (τ^2)) ((1 - 
                              2 λ) (σ^2) - 
                        nSig λ (τ^2))) (omegasq^3))/.omegasq -> (1/(6 (γ^6) (σ^6) (τ^2) ((-1 + 2 λ) (σ^2) + 
                      nSig λ (τ^2)))) (-2 (iInv^2) (γ^4) (σ^4) ((-1 + (λ^2)) (σ^4) + 
                      nSig λ (-3 + λ (5 + λ)) (σ^2) (τ^2) + 
                      3 (nSig^2) (λ^2) (τ^4)) + (2 (2^(1/3)) (iInv^4) (γ^8) (-1 + λ) (σ^10) ((-1 + λ) (((1 + λ))^2) (σ^6) - 
                          nSig λ (1 + λ) (6 + λ (-13 + 4 λ)) (σ^4) (τ^2) + 
                          (nSig^2) (λ^2) (-3 + (-6 + λ) (-4 + λ) λ) (σ^2) (τ^4) - 3 (nSig^3) (-3 + λ) (λ^3) (τ^6)))/((((iInv^6) (γ^12) (σ^14) (-2 (((-1 + (λ^2)))^3) (σ^10) + 3 nSig λ (((-1 + (λ^2)))^2) (6 + λ (-13 + 4 λ)) (σ^8) (τ^2) + 6 (nSig^2) (((-1 + λ))^2) (λ^2) (1 + λ) (6 + λ (-27 + 2 λ (8 + λ))) (σ^6) (τ^4) - (nSig^3) (((-1 + λ))^2) (λ^3) (27 + λ (171 + λ (-261 + λ (-29 + 2 λ)))) (σ^4) (τ^6) + 9 (nSig^4) (((-1 + λ))^2) (λ^4) (-9 + λ (24 + λ)) (σ^2) (τ^8) + 54 (nSig^5) (((-1 + λ))^2) (λ^5) (τ^10)) + 3 (Sqrt[3]) √((iInv^12) (nSig^2) (γ^24) (((-1 + λ))^3) (λ^4) (σ^28) (τ^4) ((((-1 + 2 λ) (σ^2) + nSig λ (τ^2)))^2) (-(-1 + λ) (((1 + λ))^4) (σ^12) + 2 nSig (((1 + λ))^3) (2 + (-3 + λ) λ (-1 + 3 λ)) (σ^10) (τ^2) - (nSig^2) (((1 + λ))^2) (-4 + λ (-34 + λ (45 + λ (77 + (-53 + λ) λ)))) (σ^8) (τ^4) - 6 (nSig^3) λ (1 + λ) (-6 + λ (-5 + λ (48 + λ (4 + (-26 + λ) λ)))) (σ^6) (τ^6) + (nSig^4) (λ^2) (81 + λ (-225 + λ (-438 + λ (294 + (169 - 9 λ) λ)))) (σ^4) (τ^8) + 8 (nSig^5) (λ^4) (-54 + 5 λ (9 + λ)) (σ^2) (τ^10) + 108 (nSig^6) (-1 + λ) (λ^4) (τ^12)))))^(1/3)) + (2^(2/3)) ((((iInv^6) (γ^12) (σ^14) (-2 (((-1 + (λ^2)))^3) (σ^10) + 3 nSig λ (((-1 + (λ^2)))^2) (6 + λ (-13 + 4 λ)) (σ^8) (τ^2) + 6 (nSig^2) (((-1 + λ))^2) (λ^2) (1 + λ) (6 + λ (-27 + 2 λ (8 + λ))) (σ^6) (τ^4) - (nSig^3) (((-1 + λ))^2) (λ^3) (27 + λ (171 + λ (-261 + λ (-29 + 2 λ)))) (σ^4) (τ^6) + 9 (nSig^4) (((-1 + λ))^2) (λ^4) (-9 + λ (24 + λ)) (σ^2) (τ^8) + 54 (nSig^5) (((-1 + λ))^2) (λ^5) (τ^10)) + 3 (Sqrt[3]) √((iInv^12) (nSig^2) (γ^24) (((-1 + λ))^3) (λ^4) (σ^28) (τ^4) ((((-1 + 2 λ) (σ^2) + nSig λ (τ^2)))^2) (-(-1 + λ) (((1 + λ))^4) (σ^12) + 2 nSig (((1 + λ))^3) (2 + (-3 + λ) λ (-1 + 3 λ)) (σ^10) (τ^2) - (nSig^2) (((1 + λ))^2) (-4 + λ (-34 + λ (45 + λ (77 + (-53 + λ) λ)))) (σ^8) (τ^4) - 6 (nSig^3) λ (1 + λ) (-6 + λ (-5 + λ (48 + λ (4 + (-26 + λ) λ)))) (σ^6) (τ^6) + (nSig^4) (λ^2) (81 + λ (-225 + λ (-438 + λ (294 + (169 - 9 λ) λ)))) (σ^4) (τ^8) + 8 (nSig^5) (λ^4) (-54 + 5 λ (9 + λ)) (σ^2) (τ^10) + 108 (nSig^6) (-1 + λ) (λ^4) (τ^12)))))^(1/3)))]

2.7  Limits for Derivation of Information Market Equilibrium

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

FullSimplify[(((τ^2) ((iInv^4) ((1 + R) (σ^2) + nSig (τ^2)) + (γ^4) (τ^4) ((σ^2) + nSig (τ^2)) (ω^4) + (iInv^2) (γ^2) (τ^2) (2 ((σ^2) + nSig (τ^2)) (ω^2) + R (σ^2) ((xbar^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]
FullSimplify[
      D[(((τ^2) ((iInv^4) ((1 + R) (σ^2) + nSig (τ^2)) + (γ^4) (τ^4) ((σ^2) + nSig (τ^2)) (omegasq^2) + (iInv^2) (γ^2) (τ^2) (2 ((σ^2) + nSig (τ^2)) omegasq + R (σ^2) ((xbar^2) + omegasq))))/(2 ((((iInv^2) ((1 + R) (σ^2) + nSig (τ^2)) + (γ^2) (τ^2) ((σ^2) + nSig (τ^2)) omegasq))^2))), omegasq] /. 
        nSig -> 0]
FullSimplify[
      Solve[Simplify[
            D[(((τ^2) ((iInv^4) ((1 + R) (σ^2) + nSig (τ^2)) + (γ^4) (τ^4) ((σ^2) + nSig (τ^2)) (omegasq^2) + (iInv^2) (γ^2) (τ^2) (2 ((σ^2) + nSig (τ^2)) omegasq + R (σ^2) ((xbar^2) + omegasq))))/(2 ((((iInv^2) ((1 + R) (σ^2) + nSig (τ^2)) + (γ^2) (τ^2) ((σ^2) + nSig (τ^2)) omegasq))^2))), omegasq] /. 
              nSig -> 0] == 0, {omegasq}]]

-(((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
FullSimplify[
      D[-(((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]]
FullSimplify[
      Solve[-(((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))) == 
          0, {λ}]]

FullSimplify[
        Limit[focEtermNewsSubscribers, 
          nSig -> (((1 - 2 λ) (σ^2))/(λ (τ^2)))]]/(1 + R)
FullSimplify[
        Limit[focVartermNewsSubscribers, 
          nSig -> (((1 - 2 λ) (σ^2))/(λ (τ^2)))]]/(1 + R)
FullSimplify[
        Limit[focDelfactorPriceWatchers, 
          nSig -> (((1 - 2 λ) (σ^2))/(λ (τ^2)))]]/(1 + R)
FullSimplify[
      Limit[(- 
                (iInv^6) (nSig^3) (λ^4) (τ^2) ((σ^2) + 
                  nSig (τ^2)) + 
            (iInv^4) nSig (γ^2) (λ^2) (σ^2) ((1 - (λ^2)) (σ^4) - 
                  3 nSig (λ^2) (σ^2) (τ^2) - 
                  3 (nSig^2) λ (τ^4)) (ω^2) + 
            (iInv^2) (γ^4) (σ^4) ((1 - (λ^2)) (σ^4) + 
                  nSig λ (3 - λ (5 + λ)) (σ^2) (τ^2) - 
                  3 (nSig^2) (λ^2) (τ^4)) (ω^4) - (γ^6) (σ^6) (τ^2) ((1 - 2 λ) (σ^2) - 
                  nSig λ (τ^2)) (ω^6)), 
        nSig -> (((1 - 2 λ) (σ^2))/(λ (τ^2)))]]