Two competing inventors, Sam and Abigale, have both independently invented a new type of highly efficient battery. They have effectively cornered the market. The world inverse demand for these new batteries is given by = 200 2 , where Q is the combined output of Sam ( ) and Abigale ( ) .

a) Sam wishes to produce the profit-maximising quantity of batteries. Given Abigale's choice of output, , write an equation for the residual demand faced by Sam. (1 mark) b) Derive Sam's residual marginal revenue curve.

c) Assume that the marginal and average total cost of producing the new batteries is $50 per unit. Derive Sam's reaction function.

d) Repeat steps (a), (b), and (c) to find Abigale's reaction function to Sam's output choice.

e) Substitute Sam's reaction function into Abigale's to solve for Abigale's profitmaximizing level of output. Then, use your answer to find Sam's profit-maximizing level of output.

f) Determine industry output, the price per unit of batteries, and the profits of both Sam and Abigale.

g) If Abigale were to quit the business because she no longer needed the money, and Sam were to become a monopolist, what would happen to industry quantity, price, and profit?