The world desperately needs a vaccine for COVID-19, so the Gates Foundation decides to offer a prize to any pharmaceutical company who can create one. Suppose there are two pharmaceutical companies that have the resources to do so.

Each pharmaceutical company must independently choose an effort levelebetween 0 and 1 to invest in vaccine research. The effort can be interpreted as the percentage of the company's resources they devote to vaccine research. Assume the company's probability of developing a vaccine is alsoe. In other words, if they invest no time, there is zero probability they discover a vaccine; if they investe= 0.5, they have a 50% probability of finding a vaccine, and so on. For each pharmaceutical companyi, the cost (in billions of dollars) of choosing effort leveleis given by

c_i(e_i) = 51e_i²

The Gates foundation decides to offer a prize ofPbillion dollars to any firm or firms that find a vaccine.If both firms are successful, they split the prize.

(a)

Suppose the Gates foundation were to offer a prize of $120 billion for a vaccine. Find the Nash equilibrium effort level exerted by each company. (That is, find thee* such that if one company were exerting efforte*, the other's best response would also be to exert efforte*.) answer to two decimal places:

(B)

Suppose the Gates foundation actually values a vaccine at $120 billion, but instead of offering a $120 billion prize, they decide to offer a prize of P billion dollars.

For the purposes of this question, let's define "social welfare" as the expected value to the Gates Foundation minus the costs to the two firms:

W(e) = 120 * Prob{vaccine found | e} - 2c(e)

How large of a prize would maximize social welfare? (That is, what prize value P* would result in the level of effort that maximizes W(e) in the equilibrium that would occur if such a prize were offered?)