A building contractor is submitting an estimate to a potential customer for carrying out some construction work at their home. The builder estimates that if he offers to carry out the work for $150,000 there is a 0.2 probability that the customer will agree to the price, a 0.5 probability that the customer will negotiate and get the offer decreased to $120,000, and a 0.3 probability that the customer will simply refuse the offer and give the job to another contractor. If the builder offers to carry out the work for $100,000, he estimates that there is a 0.3 probability the customer will accept the price, a 0.6 probability the customer will bargain for a lower price of $80,000, and a 0.1 probability the customer will refuse the deal altogether and give the job to another contractor.

a.) Determine which price the builder should quote in order to maximize the expected payment he receives from the customer.

b.) Suppose that, after some questioning, the builder is able to make the following statements:

a. I am indifferent between receiving $120k for certain or entering a lottery that will give me a 0.9 probability of winning $150k and a 0.1 probability of winning $0.

b. I am indifferent between receiving $100k for certain or entering a lottery that will give me a 0.85 probability of winning $150k and a 0.15 probability of winning $0.

c. I am indifferent between receiving $80k for certain or entering a lottery that will give me a 0.75 probability of winning $150k and a 0.25 probability of winning $0.

c.) Given the above statements, sketch the builder's utility function and comment on what it shows.

d.) In light of the above statements, which price should the builder quote to the customer, and why?