Assume there are H agents with constant absolute risk aversion $\alpha$. There is a risk-free asset, and two risky assets with distribution $S1$ ~ $N(\mu; \Sigma)$, where $\mu \in \mathbb{R}^2$ and $\mu \in \mathbb{R}^{2\times2}$. Assume $H_U$ agents are not aware of the existence of the second risky asset. If $H_U = 0$, then the equilibrium prices are $S_0^* = \frac{1}{R_f}\mu - \frac{\alpha}{HR_f}\Sigma\theta_{tot}$ where $\theta_{tot}$ is the total supply of risky assets.

Assume now $0 < H_U < H$

a) Prove that in equilibrium $(S_0)_1 = (S_0^*)_1$ and

$(S_0)_2 = (S_0^*)_2 - \frac{\alpha}{HR_f}\frac{H_U}{H-H_U}(Var((S_1)_2) - \frac{cov((S_1)_1,(S_1)_2)^2}{Var((S_1)_1)})(\theta_{tot})_2$

and hence $(S_0)_2 < (S_0^*)_2$

b) Let $R_1$ and $R_2$ be the returns of the two risky assets. Prove that there exist $A > 0$ and $\lambda$ such that

$E[R_1] = R_f +\lambda\frac{cov(R_1,R_m)}{Var(R_m)}$

$E[R_1] = A + R_f +\lambda\frac{cov(R_2,R_m)}{Var(R_m)}$

$\lambda = E[R_m] - R_f - A\frac{(S_0)_2(\theta_{tot})_2}{S_0^T\theta_{tot}}$

where $R_m$ is the return of the market portfolio.

My doubt comes early in the problem: how should I reflect the condition on $H_U$ mathematically? I understand the formula for $S_0^*$ given in the introduction, but dont know how to introduce that $H_U$ information. Thank you in advance.


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