Consider an infinite-horizon version of the model in Section 21.5 in which both investors agree the dividend process is a two-state Markov chain, with states D = 0 and D = 1. Suppose the investors’ beliefs Ph satisfy, for all t ≥ 0, P1(Dt+1 = 0|Dt = 0) = 1/2 , P1(Dt+1 = 1|Dt = 0) = 1/2 , P1(Dt+1 = 0|Dt = 1) = 2/3 , P1(Dt+1 = 1|Dt = 1) = 1/3 , P2(Dt+1 = 0|Dt = 0) = 2/3 , P2(Dt+1 = 1|Dt = 0) = 1/3 , P2(Dt+1 = 0|Dt = 1) = 1/4 , P2(Dt+1 = 1|Dt = 1) = 3/4 .

Assume the discount factor of each investor is δ = 3/4. For s = 0 and s = 1, set Vh(s) = Eh



∞

t=1

δt Dt |D0 = s



.

For each h, use the pair of equations Vh(s)

δ = Ph(Dt+1 = 0|Dt = s)Vh(0) + Ph(Dt+1 = 1|Dt = s)[1+ Vh(1)]

to calculate Vh(0) and Vh(1). Show that investor 2 has the highest fundamental value in both states [V2(0) > V1(0) andV2(1) >V1(1)] but investor 1 isthe most optimistic in state D = 0 about investor 2’s future valuation, in the sense that P1(Dt+1 = 0|Dt = 0)V2(0)+ P1(Dt+1 = 1|Dt = 0)[1+V2(1)]
> P2(Dt+1 = 0|Dt = 0)V2(0) +P2(Dt+1 = 1|Dt = 0)[1+V2(1)].