"Technical Analysis in Financial Markets"

Because of the characteristics of the demand function (6.24) for the risky asset, traders in belief group B₁ are restricted in the fraction of individual wealth they can go short, that is

a σ²

<y_t^h<∞

This implies that there is an upperbound on the fraction of total wealth traders in belief group B₂ can go long, that is

∑

h ∈ B₂

q_t^h y_t^h <1+

a σ²

∑

h ∈ B₁

q_t^h , or equivalently c₄<1+c₃

a σ²

Thus the denominator in (6.61) is positive. Now the question is whether the nominator of (6.61) is also positive, so that there is a unique positive equilibrium price. It is clear that c₁≥ 0. If the initial wealth invested in the risk free asset is positive, then according to (6.30) the total wealth at time t should be at least be equal to the value of the total number of shares: W_t=c₂+ s P_t ≥ s P_t, implying c₂ ≥ 0. Because for all beliefs h: q_t^h ≥ 0 it is also true that c₃≥ 0. c₄ ∈ IR and can be of either sign. Hence under these relationships the nominator of (6.61) is positive, because:

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

(c₁ c₃ R s-c₂ c₃R + a c₂ c₄ σ²) + (c₁ c₃ R s + c₂ c₃ R -a c₂ c₄ σ²)²= 2 c₁ c₃ R s > 0.

We have shown that the nominator and denominator of (6.61) are both positive, so that we have proven that for s>0 the model yields a unique positive equilibrium price.

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