Consider the model proposed by Kyle [1147] and described in Sect. 10.2.

(i) Suppose that the functions \(X\) and \(P\) are linear:

\[X(\tilde{d})=\alpha+\beta \tilde{d} \quad \text { and } \quad P(\tilde{x}+\tilde{z})=\mu+\lambda(\tilde{x}+\tilde{z})\]

for some constants \(\alpha, \beta, \mu\) and \(\lambda\) to be determined. Show that the profit maximization constraint (10.10) implies that

\[\beta=\frac{1}{2 \lambda} \quad \text { and } \quad \alpha=-\mu \beta\]

(ii) By relying on the result obtained in the first step, prove that the market efficiency condition (10.11) implies that

\[\lambda=\frac{\beta \sigma_{d}^{2}}{\beta^{2} \sigma_{d}^{2}+\sigma_{z}^{2}} \quad \text { and } \quad \mu-\bar{d}=-\lambda(\alpha+\beta \bar{d})\]

Deduce that Proposition 10.3 holds.