III. An upstream monopoly sells the good x to a downstream monopoly. The downstream monopoly uses this good as an input to produce its output y. The production function of the downstream monopoly is y = x. The downstream monopoly sells its output to final consumers whose aggregate demand curve is y = 12 p. The upstream monopoly's cost function is c (x)=2x.

a) . Find the quantity x the upstream monopoly sells to the downstream monopoly and the quantity y the downstream monopoly sells to the final consumers. Find also the price k the upstream monopoly charges the downstream monopoly and the price p the downstream monopoly charges the final consumers. Compute the profits of each monopoly.

b) Assume the two monopolies merge into a single monopoly that produces y from x internally (according to the same production function) and sells it directly to the final consumers. What will be the quantity y and the price p of the output? Compute the profits of the merged monopolies