Strategic Demand and Kyle’s Lambda in a CARA-Normal Market

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# How to solve for Kyle's $\lambda$ that emerges in the demand functions of the informed and uninformed traders in the $1989$ model?












I will restate here a problem that I am finding a bit difficult to solve and I have already posted here. I summarized the problem as it follows.

From Albert S. Kyle's 1989 model.

Suppose that the are trhee types of traders in the market, informed traders (I), uninformed traders (U) and noise traders. The population of $I$ traders is $N$ (namely $n=1,\dots,N$) and that of $U$ traders is $M$ (namely $m=1,\dots,M$). Both of them have CARA preferences denoted as

$$u_n(\pi_{I_n}) = -e^{-\rho_I\pi_{I_n}}, \quad \text{and} \quad u_m(\pi_{U_m}) = - e^{-\rho_U\pi_{U_m}}$$

where $\pi_{I_n} = (u-p)x_n$ and $\pi_{U_m} = (u-p)x_m$ where $\rho_I$ and $\rho_U$ stunds for the risk averstion of the informed and the uninformed traders. $u\sim N(0, \sigma_u^2)$ is the value of the risk asset, $p$ denotes its price. Each informed trader has a signal $s_n = u + e_n$ and the noise traders stochastic demand is traded as $z \sim N(0, \sigma_z^2) $. All $n+2$ random variables $u, z, e_1, \dots, e_n$ are inpepedently distributed. The author says that solving for the demand functions of the players, then

$$X_n(p, s_n)= \mu_I +\beta s_n - \gamma_I p, \quad \ Y_m(p) = \mu_U - \gamma p\tag{1}$$

and the market clearing condition of for the price is equilibrium is given by the following calculation

$$\sum_{n=1}^NX_n(p, s_n) + \sum_{m=1}^MY_m(p) - z = 0\tag{2}$$

Kyle while solving for the demand schedules of the traders for which he initially claims that they have form as in $(1)$ (which is intuitive since demands are linear and decreasing with respect to the price and the utilities are ending up to have quadratic forms because of the CARA normal preferences). He shows that there is some power called $\lambda$ that in essence moves the prices and comes from the strategic demands of the traders. After solving the problems of the informed and the uninformed traders he ends up with the following

$$x_n^* = \frac{\mathbb{E}[u|p^*, s_n]-p^*}{2\lambda_I + \rho_I\mathbb{V}ar[u|p^*, s_n]}, \quad \text{and} \quad y_m^* = \frac{\mathbb{E}[u|p^*]-p^*}{2\lambda_U + \rho_U\mathbb{V}ar[u|p^*]}\tag{3}$$

I can not understand how this $\lambda_I$ and $\lambda_U$ emerges in the above solutions. Could anyone explain to some point how do we solve for the strategic demand schedules?

What I understand is that the author takes the certainty equivalent that is of the following form

$$\mathbb{E}[(u-p)x_n|p^*, s_n]-\frac{\rho_n}{2}\mathbb{V}ar[(u-p)x_n|p^*, s_n]$$

but how does $\lambda_n$ comes in play? Ι think it should be displayed somewhere in here $\pi_{I_n} = (u-p)x_n$, but I can not understand how. If I solve the above then the demand schedule of the $n$-th informed will be like

$$x_n^* = \frac{\mathbb{E}[u|p^*, s_n]-p^*}{\rho_I\mathbb{V}ar[u|p^*, s_n]}$$

Also, Marzena Rostek uses Kyle's lambda for the strategic interaction in environments where the agents have informational advantages. I do not know if she uses the same approach to solve for the demand functions or if she uses the same technique. I refer to her article in case it helps to clarify the problem.