Question

Hula Products has reintroduced the hula hoop to the world and faces a growing demand for its product in two distinct markets: the United States and Europe. Demand in these markets is:

P_U = 20 - .1Q_U

P_E = 10 - .05Q_E

where all quantities are expressed in thousands of units (i.e. Q_U = 50 means 50 thousand units).
The company has an existing stock of 95 thousand hula hoops. How many should be sent to Europe?

Answer 1

Qu + Qe = 95
So, Qu = 95 - Qe

It maximizes profit in each market according to the rule MR = MC.

US: TRu = Pu*Qu = (20 - .1Qu)Qu = 20Qu - .1Qu²
So, MRu = d(TRu)/dQu = 20 - 2(.1Qu) = 20 - .2Qu
Now, MRu = MC = 0 gives,
20 - .2Qu = 0

Europe: TRe = Pe*Qe = (10-0.05Qe)*Qe = 10Qe - 0.05Qe²
MRe = d(TRe)/dQe = 10 - 2(0.05Qe) = 10 - 0.1Qe
Now, MRe = MC = 0 gives,
10 - 0.1Qe = 0

So, 20 - .2Qu = 10 - 0.1Qe
So, 20 - .2(95 - Qe) = 10 - 0.1Qe
So, 20 - 19 + .2Qe = 10 - 0.1Qe
So, .2Qe + .1Qe = 10 + 19 - 20
So, .3Qe = 9
So, Qe = 9/.3
So, Qe = 30 thousand