Grinch is the sole owner of a mineral water spring that costlessly burbles forth as much mineral water as Grinch cares to bottle. It costs Grinch $2 per gallon to bottle this water. The inverse demand curve 3 for Grinch's mineral water is p = 20 0.20q, where p is the price per gallon and q is the number of gallons sold.

a) (i) Write down an expression for profits as a function of q.

(ii) Find the profit-maximizing choice of q for Grinch.

b) (i) What price does Grinch get per gallon of mineral water if he produces the profit-maximizing quantity?

(ii) How much profit does he make?

c) Suppose, now, that Grinch's neighbour, Grubb finds a mineral spring that produces mineral water that is just as good as Grinch's water, but that it costs Grubb $6 a bottle to get his water out of the ground and bottle it. Total market demand for mineral water remains as before. Suppose that Grinch and Grubb each believe that each other's quantity decision is independent of his own.

(i) What is the Cournot equilibrium?

ii) What is the price in the Cournot equilibrium?