Utility Responses to Signal Acquisition under Partly Informative Price

2.1  Elasticities of Key Term Moments (Appendix F)

This section derives moments of  the Updated Variance ((((τ^i)))^2) and the Key Term ((μ^i)-RP)/((((τ^i)))^2).The moments have been reported in the notebook file infgauss1.nb and are loaded. As before, the number of signals is nSig and the number of investors iInv to prevent Mathematica from confusing these variables with internal variables. Define E[(μ - RP)/(τ^2)] as expNewsSubscribers and V[(μ - RP)/(τ^2)] as varNewssubscribers for News Subscribers, and similarly for Price Watchers. So, moments of the key terms are

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))));

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)));

elastexpNewsSubscribers = 
      FullSimplify[(D[expNewsSubscribers, nSig])/(expNewsSubscribers/
              nSig)]
elastvarNewsSubscribers = 
      FullSimplify[(D[varNewsSubscribers, nSig])/(varNewsSubscribers/
              nSig)]
elasttauNewsSubscribers = 
      FullSimplify[(D[tausqNewsSubscribers, 
              nSig])/(tausqNewsSubscribers/nSig)]

elastexpPriceWatchers = 
      FullSimplify[(D[expPriceWatchers, nSig])/(expPriceWatchers/
              nSig)]
elastvarPriceWatchers = 
      FullSimplify[(D[varPriceWatchers, nSig])/(varPriceWatchers/
              nSig)]
elasttauPriceWatchers = 
      FullSimplify[(D[tausqPriceWatchers, nSig])/(tausqPriceWatchers/
              nSig)]

2.2  Derivation of Terms in Table 1 (Elasticities of Normalized Excess Return Moments, Appendix G)

To derive the terms in Table 1,  combine the previous derivations accordingly. Results are

elastexpPlainNewsSubscribers = 
      FullSimplify[(D[(expNewsSubscribers * tausqNewsSubscribers), 
              nSig])/((expNewsSubscribers * tausqNewsSubscribers)/
              nSig)]
elastexpNormalizedNewsSubscribers = 
      FullSimplify[(1/2) elasttauNewsSubscribers + 
          elastexpNewsSubscribers]
elastvarNormalizedNewsSubscribers = 
      FullSimplify[(1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers]

elastexpPlainPriceWatchers = 
      FullSimplify[(D[(expPriceWatchers  * tausqPriceWatchers), 
              nSig])/((expPriceWatchers  * tausqPriceWatchers)/
              nSig)]
elastexpNormalizedPriceWatchers = 
      FullSimplify[(1/2) elasttauPriceWatchers + 
          elastexpPriceWatchers]
elastvarNormalizedPriceWatchers = 
      FullSimplify[(1/2) elasttauPriceWatchers + (1/2) 
            elastvarPriceWatchers]

To compare the terms in the middle column of Table 1 between News Subscribers and Price Watchers, subtract them:

FullSimplify[
      elastexpPlainNewsSubscribers - 
        elastexpPlainPriceWatchers]
FullSimplify[
      elasttauNewsSubscribers - 
        elasttauPriceWatchers]
FullSimplify[
      elastexpNormalizedNewsSubscribers - 
        elastexpNormalizedPriceWatchers]
FullSimplify[
      elastvarNormalizedNewsSubscribers - 
        elastvarNormalizedPriceWatchers]

2.3  Evaluation of Terms in Table 1 for News Subscribers (Elasticities of Normalized Excess Return Moments, Appendix G)

Evaluate the pre-posterior mean and variance of the normalized expected excess return for News Subscribers for (ω^2) (omegasq) and (τ^2) (tausq).

The cutoff asset supply noise (ω^2), denoted omegasq, at which the mean normalized expected excess return elasticity for News Subscribers (1/2) elasttauNewsSubscribers + elastexpNewsSubscribers turns positive is

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

Note that the pre-posterior mean normalized expected excess return elasticity for News Subscribers monotonically increases in asset supply noise (ω^2), denoted omegasq, if (ω^2) is sufficiently large:

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

The cutoff asset supply noise (ω^2), denoted omegasq, at which the variance of the normalized expected excess return elasticity for News Subscribers (1/2) elasttauNewsSubscribers + (1/2) 
          elastvarNewsSubscribers turns positive is

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

The cutoff of the prior payoff variance (τ^2), denoted tausq, at which the pre-posterior variance of the normalized expected excess return (1/2) elasttauNewsSubscribers + (1/2) 
          elastvarNewsSubscribers turns positive is

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

Also note that the pre-posterior variance of the normalized expected excess return for News Subscribers monotonically decreases in the prior payoff variance (τ^2), denoted tausq. The argument takes two steps. Start by taking the derivative of the last term in the solution to ((1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers) with respect to tausq:

FullSimplify[
      D[((γ^2) (1 - λ) (σ^4) (ω^2) ((iInv^4) nSig (λ^2) (1 + λ) ((σ^2) + 
                        nSig tausq) ((σ^2) + 2 nSig tausq) + 
                  (iInv^2) (γ^2) (σ^2) ((1 + λ) (σ^4) + 
                        nSig (2 + λ (5 + λ)) (σ^2) tausq + 6 (nSig^2) λ (tausq^2)) (ω^2) + 
                  2 (γ^4) (σ^4) tausq ((σ^2) + 
                        nSig tausq) (ω^4)))/(((iInv^2) nSig (λ^2) ((σ^2) + 
                        nSig tausq) + (γ^2) (σ^2) ((σ^2) + 
                        nSig λ tausq) (ω^2)) ((iInv^4) (nSig^2) (λ^2) ((σ^2) + nSig tausq) + 
                  (iInv^2) nSig (γ^2) (σ^2) ((1 + (λ^2)) (σ^2) + 
                        2 nSig λ tausq) (ω^2) + (γ^4) (σ^4) ((σ^2) + nSig tausq) (ω^4))), 
        tausq]]

Then subtract the denominator from the numerator to infer whether the term changes less or more than -1 in tausq.

FullSimplify[-(((((iInv^2) nSig (λ^2) (nSig tausq + (σ^2)) + (γ^2) (σ^2) (nSig tausq λ + (σ^2)) (ω^2)))^2) ((((iInv^4) (nSig^2) (λ^2) (nSig tausq + (σ^2)) + (iInv^2) nSig (γ^2) (σ^2) (2 nSig tausq λ + (1 + (λ^2)) (σ^2)) (ω^2) + (γ^4) (σ^4) (nSig tausq + (σ^2)) (ω^4)))^2)) + (γ^2) (1 - λ) (σ^6) (ω^2) ((iInv^10) (nSig^5) (λ^6) (1 + λ) (((nSig tausq + (σ^2)))^2) + 
              2 (iInv^8) (nSig^4) (γ^2) (λ^4) (σ^2) (nSig tausq + (σ^2)) (nSig tausq (1 + λ (4 + (-1 + λ) λ)) + (2 + 
                          2 λ + (λ^3)) (σ^2)) (ω^2) + 
              (iInv^6) (nSig^3) (γ^4) (λ^2) (σ^4) (2 (nSig^2) (tausq^2) λ (6 + 3 λ + (λ^3)) + 
                    2 nSig tausq (2 + 11 λ + 
                          7 (λ^3)) (σ^2) + (5 + λ (5 + λ (3 + 7 λ))) (σ^4)) (ω^4) + 
              2 (iInv^4) (nSig^2) (γ^6) (σ^6) ((nSig^2) (tausq^2) (λ^2) (9 + λ (-1 + 2 λ)) + 
                    nSig tausq λ (6 + λ (3 + λ (10 + λ))) (σ^2) + (1 + λ) (1 + (λ^2) (3 + λ)) (σ^4)) (ω^6) + 
              (iInv^2) nSig (γ^8) (σ^8) ((nSig^2) (tausq^2) λ (10 + (-1 + λ) λ) + 
                    2 nSig tausq (1 + λ) (2 + 
                          3 λ) (σ^2) + (3 + λ (3 + 4 λ)) (σ^4)) (ω^8) + 
              2 (γ^10) (σ^10) (((nSig tausq + (σ^2)))^2) (ω^10))]

So, the  the last term in the solution to ((1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers) changes less strongly than -1 and the overall effect of tausq on ((1/2) elasttauNewsSubscribers + (1/2) 
            elastvarNewsSubscribers) is negative.

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

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)]

For News Subscribers, focEtermNewsSubscribers is strictly positive only if  (ω^2) (omegasq) exceeds the cutoff

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

For News Subscribers, focVartermNewsSubscribers is strictly positive if  (τ^2) (tausq) drops below a cutoff

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

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 evaluates to

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

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}]]

2.6  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 + (1/2) 
            elastvarNewsSubscribers, 
        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 -> ∞]]

A number of properties are worth exploring: (i) the signs of (-1 + (λ^2)) (σ^4) + 
        3 nSig (λ^2) (σ^2) (τ^2) + 
        3 (nSig^2) λ (τ^4) and \(\((\(-1\) + λ\^2)\)\ σ\^4 - 
        nSig\ λ\ \((3 - λ\ \((5 + λ)\))\)\ σ\^2\ τ\^2 + 3\ nSig\^2\ λ\^2\ τ\^4\) in the tausq limits, (ii) the sum of the utility terms in the gamma limit.

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

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)))]