Examples and exercises on a price-discriminating monopolist

In market 1 we have Q_d(p) = 10

p/2 while in market 2 we have Q_d(p) = 32

2p. The monopolist's total cost function is TC(y) = y². What outputs does the monopolist sell in each market?

We have

TR₁(y₁) = y₁(20 2y₁) TR₂(y₂) = y₂(16 y₂/2)

and hence

MR₁(y₁) = 20 4y₁ MR₂(y₂) = 16 y₂.

Thus the condition MR₁(y₁*) = MR₂(y₂*) = MC(y₁* + y₂*) is equivalent to

20 4y₁* = 16 y₂* = 2(y₁* + y₂*).

Isolating y₂* in the first equation we obtain

y₂* = 4y₁* 4.

Plugging this into the expression 20

4y₁* = 2(y₁* + y₂*) we obtain

20 4y₁* = 2(y₁* + 4y₁* 4)

28 = 14y₁*

y₁* = 2,

so that

y₂* = 4.

(I am assuming that the first-order conditions identify a maximum rather than a minimum.)