A contractor is submitting an offer to a potential customer for a consulting project. Based on previous experience, the contractor assesses that if the project is offered at $$150,000,$ then there is a $20\%$ chance that the customer would agree to the price. There is also a $50\%$ chance that a price of $$120,000$ would be accepted and a $30\%$ chance that the customer would simply refuse the offer. On the other hand, if the contractor offers the project for $$100,000,$ then there is a $30\%$ chance that the customer would accept the offer, $60\%$ chance that the customer would eventually negotiate down to $$80,000$ and $10\%$ chance that the customer would refuse the offer.$$

a$.$ Which price should the contractor offer to the potential customer?

b$.$ Suppose that the contractor$$s preference is revealed through a series of questions:

$-$ The contractor is indifferent between receiving $$120,000$ for certain or entering a lottery that will give them a $90\%$ probability of winning $$150,000$ and $10\%$ probability of winning $$0.$

$-$ The contractor is indifferent between receiving $$100,000$ for certain or entering a lottery that will give them a $85\%$ chance of winning $$150,000$ and $15\%$ probability of winning $$0.$

$-$ The contractor is indifferent between receiving $$80,000$ for certain or entering a lottery that will give them a $75\%$ chance of winning $$150,000$ and $25\%$ probability of winning $$0.$

Plot the contractor$$s utility function and identify their attitude towards risk.

c$.$ Based on the contractor's preference, what price should they offer to the customer?