Honor Code: You must do these problems on your own, and submit your answers no later than 11 PM on Sunday, November 11.
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Leon, the lion tamer, has decided to try adding a new lion to his pride. He's visited a commercial lion-breeding farm, where he has been offered a cub for $6,000. From past experience, Leon believes that there's a 60% chance that a lion cub purchased from the farm will prove to be trainable.
It takes about six months of work to really learn for certain whether a lion is trainable or not. If the lion is trainable, Leon feels that his long-term gain, netted against the cost of caring for the lion, is $15,000. (That is, he'll net $9,000 in profit after also figuring in the cost of purchase.) However, if the lion proves to be untrainable, his only option is to donate the lion to a local zoo for a charitable-deduction tax break. After netting maintenance expenses against the deduction, he still comes out $2,000 behind (for an overall loss of $8,000).
What is Leon's expected profit (taking both the purchase price and long-term costs and gains into account) if he buys a cub from the farm? $
Leon knows that, after working with a lion cub for a week, he'll get a "feeling" for whether the lion will eventually prove to be trainable: If a lion really is trainable, he'll have a 90% chance of feeling optimistic (and a 10% chance of feeling pessimistic) at the end of the week; if a lion really is untrainable, he�ll have only a 50% chance of feeling optimistic. (Leon, like any good lion tamer, leans towards optimism.)
The farm has offered Leon a special deal: For an extra (nonrefundable) $500, the cub will be returnable up to a week after purchase for a full refund of the base purchase price. (That is, Leon pays $6,500 up front, but gets $6,000 back if he returns the cub after a week.) Leon considers the cost of caring for (and working with) a lion cub for just a week to be negligible.
What is Leon's optimal strategy?
Following his optimal strategy, what is Leon's expected overall expected profit? $
If the cost of the option were up for negotiation (instead of being $500), what is the most Leon should be willing to pay for the option? $
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