I really need help with these questions:

Thank you

Question 1 Consider a continuum of firms seeking developing a Corona virus antibody test. The firms have different prospects for their test's effectiveness. The prospective effectiveness of a firm's test is denoted by 0, with 0 s 0 s 1. The poorest prospective effectiveness is denoted by 0, the best by 1. We refer to individual firms by their unique value of 0. The firms' 0 are uniformly (i.e. 'evenly") distributed over the interval [0, 1]. The actual prospective effectiveness of a firm's test is only known to the firm. A firm can either sell the rights to its test before it is fully developed to one of many (possibly international) health agencies at a price of P, or market the test themselves after it is fully developed and its effectiveness is known. Marketing a test of prospective effectiveness 0 when it is fully developed gives the firm an expected revenue of: M(0) = 1 + 50. There are as many identical health agencies potentially willing to buy the rights to a test as there are firms, so the health agencies do not derive economic rents from buying tests early. The health agencies have access to unlimited financial resources. A health agency can only buy the rights to one single test before it is developed. If the firm's test is of prospective effectiveness e and the price is P, the health agency obtains a utility of we, P) :=2+ 0 -P. Not buying the rights to any test gives the health agency a utility of 0. For any price P each health agency knows the average prospective effectiveness F of the not fully developed tests offered in the rights market at that price. But only after the sale of the rights to a test has been completed, the actual prospective effectiveness of the test becomes known to the firm. Sellers (Firms): a) Denote by V(P) the firms who are willing to sell the rights to their not fully developed test at a price P. For P = 3 determine V(P) graphically in a diagram. 10 marks b) Denote by 0"(P) the 'critical' or "watershed' firm which at price Pis indifferent between selling the rights to its test before it is fully developed and waiting to market its test after it has been fully developed. Calculate gail(P). Determine the supply function S(P) of rights to not fully developed tests. Depict the supply function S(P) in a diagram. 15 marks Continued...c) Let F(P) denote the average prospective effectiveness of not fully developed tests offered to the rights market at price P. Determine F(P). Depict F(P) in a diagram. 10 marks Buyers (Health agencies): d) At which combinations (F, P) of average prospective effectiveness F of not fully developed test offered to the rights market, and of price P, will health agencies be willing to buy the rights to not fully developed tests? At which combinations (F, P) will they refuse to buy? When will they be indifferent between buying and not buying? Explain. 10 marks e) Let D(P; F) denote the demand for the rights to not fully developed tests at price P, given that the average prospective effectiveness of the tests offered to the rights market is F. Depict the demand D(P, F) for F = 0.6 in the diagram of the supply function S(P) in part b). At which price does demand equal supply for F = 0.6? Explain. 10 marks f) Calculate D(P, F). At which combinations for P and F will the market for rights to not fully developed tests clear? Depict these combinations in the diagram of part c). 10 marks Equilibrium: g) What is the firms' 'supply side' equilibrium condition on the combinations of F and P? What is the health agencies' 'demand side' equilibrium condition on the combinations of F and P? 10 marks h) Determine the equilibrium combination (F, P) graphically in the diagram of part c). Calculate the equilibrium combination (F, P). 10 marks Continued...Interpretation: Assume that health agencies start to worry that there may be a shortage of test kits at a later stage, and therefore become more eager to secure the rights to a test before it is fully developed. Suppose that as a result their utility function changes from u(0, P) := 2 + 0-P to u(0, P) := 2 + 1.250- P. i) Adapting the above graphical analysis in a new diagram, how do you expect this change to affect the equilibrium price P and the average prospective effectiveness F in the market for rights to not fully developed test? How do you expect this change to affect the conditions in the market for test kits after the all tests have been developed? Discuss. 15 marks