Suppose there was just one company working on developing a COVID vaccine. We'll count money in millions of dollars ($ MM). Let x be the amount of money (in millions) the firm invests up front in developing a vaccine, so x = 1 means the firm invests a million dollars ($1 MM), and x = 1,000 means the firm invests a billion dollars (a thousand million, or $1,000 MM). And suppose that for any amount of money x the company invests, the probability of successfully developing a vaccine is 1 - 1/x. (So a firm that invests $1 MM has no chance of developing a vaccine, but a firm that invests $10 MM is 90% likely to develop one.)

Suppose that a successful vaccine would earn the company ten billion dollars, or $10,000 MM; while the value to the world as a whole would be nine trillion dollars, or $9,000,000 MM (about one-fifth the US's annual GDP).

(a) Write the firm's expected profit, as a function of its investment choice x. (This is the probability of developing a vaccine times its profitability, minus the investment made. Be sure that all monetary amounts are written in $MM.)

(b) How much money would the firm invest to maximize its expected profits?

(c) What's an expression for the expected total surplus, as a function of x. What is the efficient amount for the firm to invest?

(d) Suppose the US government promised the firm an additional prize of one hundred fifty billion dollars ($150,000 MM), on top of the profit they were already going to earn, if they successfully develop the vaccine. How much would the firm now choose to invest? Would this increase or decrease social surplus? By how much?