Question $1.$ Sensitivity analysis for the Very Simple Example. $(20$ points.$)$

The full fare is $$100,$ the discounted fare is $$70.$ In the next period $($the last one before departure$)$

there will be a request for the discounted fare with probability p$\_($d$),$ and a full fare request with

probability p$\_($f$)$; probability of no request is $1-$p$\_($d$)-$p$\_($f$).$

$($a$)$ What would the probability of a discount request have to be in the next period in order to

accept a discount request in this period?

$($b$)$ Assume p$\_($d$)=0\mathrm{.}5$ and p$\_($f$)=0\mathrm{.}4.$ How much should the discount price increase in order to accept it

in this period?

$($c$)$ Can you find probabilities p$\_($d$)$ and p$\_($f$)$ for the example that will make you reject the full fare this

period?

$($d$)$ Find examples of class request probabilities p$\_($d$)$ and p$\_($f$)$ so that the expected value of the seat in

the next period is $$85.$