Consider the model proposed in Vives [1628] and presented in Sect. 10.2. In this exercise, we are going to prove the existence of a unique linear symmetric equilibrium of the economy, characterized as in Proposition 10.6.

(i) As usual, start from the conjecture that there exists a linear equilibrium. A first consequence of the linearity of the equilibrium, together with the distributional assumptions of the model, is that the equilibrium price will be normally distributed. Consider then the expected utility maximization problem of an arbitrary informed trader \(i \in[0,1]\). Show that, conditionally on the observation of the realization \(p\) of the equilibrium price \(\tilde{p}\) and of the realization \(y_{i}\) of the private signal \(\tilde{y}_{i}\), the demand schedule of the \(i\)-th informed trader satisfies

\[X_{i}\left(p, y_{i}\right)=\frac{\mathbb{E}\left[\tilde{d} \mid p, y_{i}\right]-p}{a \operatorname{Var}\left(\tilde{d} \mid p, y_{i}\right)}\]

(ii) For any price \(p\), let \(L(p):=\int_{0}^{1} X_{i}\left(p, y_{i}\right) \mathrm{d} i+\tilde{z}\) be the aggregate demand in correspondence of the realization \(y_{i}\) of the private signal \(\tilde{y}_{i}\). Suppose that

\[X_{i}\left(p, y_{i}\right)=\alpha y_{i}+b p+c, \quad \text { for every } i \in[0,1]\]

for some constants \(\alpha, b\) and \(c\), for every realization \(y_{i}\). Show that the equilibrium price set by the risk neutral market maker via the rule \(\tilde{p}=\mathbb{E}[\tilde{d} \mid L(\cdot)]\) is given by

\[\tilde{p}=(1-\lambda \alpha) \bar{d}+\lambda(\alpha \tilde{d}+\tilde{z})\]

where \(\lambda=\alpha \tau_{z} /\left(\alpha^{2} \tau_{z}+\tau_{d}\right)\).

(iii) By making use of the results established in the two previous steps of the exercise, deduce that the optimal demand schedules of the informed traders are given as in (10.18), so that Proposition 10.6 holds.