The small nation of Westeros has population 10,000, and none of its citizens has health insurance. Vandelay Industries (VI), a successful importer/exporter, sees an opportunity to branch into the health-care business. Currently, to our great mathematical convenience, the citi- zens' annual health care costs are uniformly distributed between 0 and $10,000. Each citizen's cost of care is known to them, and is the same every year, but is private. VI could care for any customer more effi- ciently; an individual whose care currently costs c can be provided the same care by VI for only .8c. The market will work as follows: VI chooses a price p and will offer full care for the next year to anyone who signs up at this price. Citizens know their own cost c and sign up if p < c. (a) For a given price p, how many customers will sign up? (b) What will the average c be for these customers, and what will Vandelay's average cost and total costs be? (c) What is Vandelay's profit as a function of p, and what is the profit-maximizing p?