Kyle: commitment does not help

In the one-period Kyle model, suppose the insider could publicly commit to a linear strategy $x = b(v - p_0)$ before trading, and the market maker then prices competitively given $b$. (a) Show that the market maker's $\lambda(b) = b\Sigma_0/(b^2\Sigma_0 + \sigma_u^2)$ and that the insider's expected profit is $\pi(b) = b\Sigma_0\sigma_u^2/(b^2\Sigma_0 + \sigma_u^2)$. (b) Maximise $\pi(b)$ and show that the optimum is the equilibrium $\beta = \sigma_u/\sigma_v$. Why is this not obvious in advance? (c) Compute $\operatorname{Var}(v \mid y)$ as a function of $b$ and show that informativeness rises with $b$.