[MUSIC] So let's go ahead and prove that combining resources is actually a good thing. We've drawn the two production possibility frontier. Here's Abe's, we chose a point on the production possibility frontier that Abe wants, which is 10 units of home goods and 56 units of market goods. We've drawn the combined production possibility frontier, and we've seen that if they completely specialize, then Sarah will be producing 60 units of home goods and Abe will be producing 96 units of market goods, and the last step is to show that if they, on the combined production possibility frontier, Sarah can magically achieve something that was impossible to achieve before. So let's do that last step. And let's calculate what Sarah will be able to consume if they combine their resources. So, if they combine their resources, they could be at any point on the production possibility frontier. But let's just assume, for the sake of this proof, that the point that they choose is this point, the point of full specialization. And at this point, Abe is making 96 units of market goods. And Sarah is producing 60 units of home goods. So, what is Sarah left with at the end of the day? What is Sarah's share, with these combined resources that they are producing? Well, let's look what happens in terms of market goods. We know that Abe is producing 96 units of market goods, out of which he will eating 56, so that leaves Sarah with 40 units of market goods. What about home goods? Well Sarah's making the home goods, and she's making 60 units of home goods. But of course, she has to give Abe his share, which is 10 units. And that leaves her 50 units of home goods. So at the end of the day, Abe, again, is on his production possibility frontier and this is the share that Sarah ends up with. Now my claim is, is that Sarah is better off than she could possibly be if she were producing by herself, and we can go ahead and prove that. Let's try and figure out how many hours it would take Sarah to produce this bundle by herself. To produce their market goods, 40 units of market goods at her rate, which is 10 units per hour, this would take her 4 hours. And what about the home goods? 50 units of home goods at 5 units per hour, this would take her 10 hours. And we add this together, and we can see that it would take Sarah 14 hours to achieve this share. Another way of showing that is finding this point of 40 units of market goods and 50 units of home goods on the production possibility frontier. 50 units of home goods is just about here. 40 units of market goods is just about there. And lo and behold, this point is this point here, and clearly, this is outside Sarah's production possibility frontier. In other words, what we've done is the following. We've proven that by combining their resources, Abe and Sarah can achieve the impossible. Abe achieves exactly what he could before, but Sarah achieves something that was not possible before. In other words, we've created something from nothing.