Consider a firm that is investing in research to treat a particular disease.For simplicity, assume that the firm is allowed to sell its invention for one period as a monopolist.Subsequently, the market becomes competitive. The probability that the firm will successfully discover a treatment is equal to, which rises with research investment.That is, the more the firm invests, the more likely it is to discover the treatment.Before the firm has decided how much to invest, its expected profits are equal to, where is the total amount of profit it earns in the event that it discovers and launches the new treatment, and where is the cost of capital.If the firm succeeds in discovering the drug, its marginal cost will be given by, and the (inverse) demand for the drug is given by.The latter condition also implies that the willingness to pay among consumers for the Qth unit of the drug is equal to.This drug's price will be set via Nash-bargaining between the drug company and a single insurer that covers all consumers.

1. Joint surplus is maximized when the quantity of the drug sold sets the marginal cost of production equal to the marginal consumer's willingness to pay.In light of this fact, what is the joint surplus-maximizing quantity of the drug sold in the event of discovery?What is the total willingness to pay for the drug among consumers?(Hint:What is the relationship between the inverse demand function and consumer willingness to pay?)
2. Assume that the insurance company can extract from consumers their total willingness to pay in the form of additional health insurance premiums.

• Define as total willingness to pay for the quantity.Defineas the price per unit that is paid to the manufacturer.With these definitions in hand, write down an expression for the insurer's profit, as a function of.Under this assumption, write down an expression for the profits of the insurer, as a function of quantity sold to consumers?
• Write down an expression for the profits of the drug company, as a function of quantity sold to consumer.
• Using your answers to parts i) and ii), derive an expression for joint surplus as a function of .
• Using the joint surplus function in part iii) and your findings from part a), calculate in dollar terms the equilibrium joint surplus that results from Nash-bargaining between the drug company and the insurer

1. Suppose that the Nash-bargaining weights are equal across the drug company and the insurer.What will its total profits from discovery be?What does this imply for the price paid to the manufacturer?
2. Now suppose that the federal government implements price controls for pharmaceuticals.A new law stipulates that price received by the manufacturer cannot exceed $30.Will this change the quantity of drugs sold?Now what are the firm's total profits from discovery?
3. According to the Nordhaus model of innovation, what will happen to innovation investments as a result of the government's new price control strategy?