"Technical Analysis in Financial Markets"

If we divide (6.59) by (6.60) we find that the fraction of total wealth invested according to belief h at time t converges to:

W_t^h

W_t

→

q_t^h.

C. Equilibrium price for s>0

If s>0, then the derivation of the equilibrium price becomes more complex. By substituting (6.24) and (6.29) in (6.35) we can rewrite (6.35) as the solution to a quadratic equation of P_t. The formulas for the equilibrium price P_t are:

c₁=

∑

h ∈ B₁

q_t^h

c₃

E_t^h(P_t+1+D_t+1);

c₂=R(W_t-1-s P_t-1)+s D_t;

c₃=

∑

h ∈ B₁

q_t^h;

c₄=

∑

h ∈ B₂

q_t^h y_t^h;

Discr=(c₁ c₃ R s + c₂ c₃ R - a c₂ c₄ σ²)² + 4 a c₁ c₂ c₃ R s σ²;

P_t=

(c₁ c₃ R s-c₂ c₃R + a c₂ c₄ σ²) + Discr

2 s (c₃ R + a (1-c₄) σ²)

(61)

Here c₁ is the net present value of the average of the expected future price plus dividend by all agents in belief group B₁, c₂ is the total amount of money invested in the risk free asset by all agents, c₃ is the total fraction of market wealth assigned to beliefs in group B₁ and c₄ is the fraction of market wealth invested in the risky asset by agents in belief group B₂ at time t. For the equilibrium equation to be solvable for P_t it is necessary that there is a belief h ∈ B₁ for which q_t^h>0. If for all beliefs h ∈ B₁: q_t^h=0, then there is no solution for P_t. Further, an upperbound should be imposed on the fraction of total market wealth traders in group B₂ can go long in the market. If s>0, then the fraction of total market wealth invested in the risky asset lies between 0 and 1, that is

0 ≤

∑

h ∈ B₁

q_t^h y_t^h +

∑

h ∈ B₂

q_t^h y_t^h < 1,

or equivalently

∑

h ∈ B₁

q_t^h y_t^h ≤

∑

h ∈ B₂

q_t^h y_t^h < 1-

∑

h ∈ B₁

q_t^h y_t^h.

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