Kyle model. There is a stock with true value $V,$ which is revealed to all at the end of trade. It is known by all that $V$ is drawn from a distribution that has a mean of zero $($that is$,p_0$ is zero$)$ and a variance of ${}_0=10\mathrm{.}7.$ There are three agent types:

$($i$)$ a single informed trader who knows $V$;

$($ii$)$ many uninformed but rational market makers who compete with each other;

$($iii$)$ many noise traders who submit orders totaling $u$ that are Gaussian with mean zero and variance ${}_u^2=11\mathrm{.}9.$

Total order flow $y$ is comprised of the informed trader's trade $x$ and the noise trade $u$ :

$y=x+u$

$($a$)$ Write down the profit function of the informed trader.

$($b$)$ The market makers choose a price impact factor $$ and set price equal

to

$p=y.$

The market maker knows that the informed trader uses a linear strat$-$

egy, that is$,$

$x=V$

The market makers compete with each other. Because there is com$-$

petition, the market makers choose $$ so that it is the best prediction

of the true value $V.$ What is the market maker's optimal $?$ For full

credit, show how $$ follows from projection.

$($c$)$ Write down the profit function for the informed trader using the

pricing rule $p=y$ and state the optimal $.$ Interpret.

$($d$)$ Solve for $,$ and also expected profit in terms of the fundamental

variances ${}_u^2$ and $.$

$($e$)$ How is liquidity expressed in the model?