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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 , with 0 1. The poorest prospective effectiveness is denoted by 0, the best by 1. We refer to individual firms by their unique value of . The firms' 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 when it is fully developed gives the firm an expected revenue of:

M() = 1 + 5.

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 and the price is P, the health agency obtains a utility of

u(, P) :=2 + - 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.

Question:

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.

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