The owner of Gator Airlines wishes to take her firm public by selling $6$ million shares. The underwriter determines that the true value will be $$15$ with probability $\mathrm{.}4$ and $$8$ with probability $\mathrm{.}6.$ There are a group of uninformed investors who are willing to buy $6$ million shares as long they expect not to lose money on average. They also assess the probability of the true value of the shares being $$15$ with probability $\mathrm{.}4$ and $$8$ with probability $\mathrm{.}6.$ There is a group of informed investors who always know whether the true value is $$15$ or $$8.$ They are willing to order $2$ million shares if the offer price is less than or equal to what they know the true value of the shares to be$.$ Assume initially that if more shares are ordered than are available for sale, the underwriter allocates the available shares on a pro$-$rata basis based on the size of each groups order $($i$.$e$.,$ the fraction of shares a group gets is equal to: $($the group's order$)/($total shares ordered$).$

$($a$)$ Calculate the highest offer price at which the entire issue will surely sell.

$($b$)$ Could an uninformed investor make money on average by submitting an order to buy some shares of the new issue? Why or why not?

$($c$)$ If the stock is sold at the offer price you calculated in $($a$),$ what will be the expected change in the stock price on its first day of trading?

$($d$)$ Assume now that when an issue is oversubscribed the underwriter follows a rule of allocating $3/4$ of the shares to the informed investors and $1/4$ of the shares to the uninformed investors. Given this new allocation rule, calculate the highest offer price at which the entire issue will always sell. Has underpricing $($i$.$e$.,$ the expected first day change in the stock price$)$ gotten larger or smaller with this new allocation rule?