Question 210 hula products has reintroduced the hula hoop to...

Question 2.10 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: United States: P(U) = 20 - 0.1Q(U) Europe: P(E) = 10 - 0.05Q(E) where all quantities are expressed in thousands of units (i.e. Q(U) = 10 should be interpreted as 10 thousand hula hoops). The company has a marginal cost of two dollars per unit in the United States (MC(U) = $2) and four dollars per unit in Europe (MC(E) = $4). Finally, the firm has a production limitation or constraint of 96 units (96 thousand hula hoops). How many hoops should be sold in the United States?

Transcript

00:01 So here the company wants to maximize profit.

00:03 So what we need to do is set up a profit function that reflects these constraints, right? so profit here is going to be equal to the u .s.

00:14 Profit, right, which is the price in the u .s., the quantity in the u .s., minus the costs in the u .s., plus the price in europe, the quantity in europe, minus the costs in europe, right? and now this is a whole bunch of variables.

00:29 I want to rewrite this in terms of one variable.

00:32 So the profit here would be equal to 20 minus 0 .1 qu times qu minus the marginal cost, right? so the marginal cost in the united states is 2.

00:47 So this is minus 2qu plus now this is the profit in europe, 10 minus 0 .05 qe.

00:59 Qe minus 4 qe.

01:03 But now we still have two variables.

01:05 But the key thing here that we know is we know that qu plus qe has to be less than or equal to 96, right? that's the binding constraint here, that these things have to be less than or equal to 96.

01:23 So there are two possibilities here, right? one, the constraint binds or two, the constraint doesn't bind.

01:30 We're going to do it with assuming that the constraint binds, right? if this, i will check that at the end.

01:39 But if this constraint doesn't bind, then we don't need the constraint at all, right? because if you want to produce 40 in europe and 30 in the united states, the constraint is totally irrelevant.

01:48 I assume that we're given this production limitation because it's actually relevant.

01:53 And so this means i can rewrite it in terms of qu is equal to 96 minus qe.

02:00 And that gives me a profit function of, if i sum all this in, 20 minus 0 .1, 96 minus qe, right? 96 minus qe minus 2 outside of 96 minus qe plus 10 minus 0 .05 qe outside of 96 minus qe plus 10 minus 0 .05 minus 4 qe, right? so now we've written it in terms of one variable.

02:37 So now i need to differentiate profit with respect to qe.

02:41 And this is a little bit tricky because i don't want to factor all this out.

02:45 But here we go nonetheless.

02:48 So this is equal to 20 minus 0 .1 outside of 96 minus qe.

02:58 And this all has a minus in front of it, right? because i'm doing the product rule on the first term...

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