Mictel Corporation has a world monopoly on the production
of personal computers. It can make two kinds of computers: low end and high
end. One-fifth of the potential buyers are casual users, and the rest are
intensive users. The costs of production of the two kinds of machines as well
as the benefits gained from the two, by the two types of prospective buyers,
are given in the following table (all figures are in thousands of dollars):
Each type of buyer calculates the net payoff (benefit
minus price) that he would get from each type of machine and buys the type that
would give the higher net payoff, provided that this payoff is nonnegative. If
both types give equal, nonnegative net payoffs for a buyer, he goes for the
high end; if both types have negative net payoff for a buyer, he does not
purchase. Mictel wants to maximize its expected profit.
(a) If Mictel were omniscient, then, when a prospective
customer came along, knowing his type, the company could offer to sell him just
one type of machine at a stated price, on a take-it-or-leave-it basis. What
machine would Mictel offer, and at what price, to what buyer? In fact, Mictel
does not know the type of any particular buyer. It just makes its catalog
available for all buyers to choose from.
(b) First, suppose the company produces just the low-end
machines and sells them for price x. What value of x will maximize its profit?
(c) Next, suppose Mictel produces just the high-end
machines and sells them for price y. What value of y will maximize its profit?
(d) Finally, suppose the company produces both types of
machines, selling the low-end ones for price x and the high-end ones for price
y. What incentive-compatibility constraints on x and y must the company satisfy
if it wants the casual users to buy the low-end machines and the intensive
users to buy the high-end machines?
(e) What participation constraints must x and y satisfy
for the casual users to be willing to buy the low-end machines and for the
intensive users to be willing to buy the high-end machines?
(f) Given the constraints in parts (d) and (e), what
values of x and y will maximize the expected profit when the company sells both
types of machines? What is the company’s expected profit from this policy?
(g) Putting it all together, decide what production and
pricing policy the company should pursue.