# Equilibrium with H agents when some of them are not aware of some assets

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.