The shape of the production possibilities curve (PPC) is caused by the law of increasing opportunity costs. If opportunity costs did not increase, PPCs would be straight lines.
The law of increasing opportunity costs says that, as you produce more of one good or service, you have to give up more of another good or service. In other words, when you start to increase production of Good A, you do not have to give up that much of Good B for every new unit of Good A that you produce. However, as you continue to increase production, your opportunity costs increase and you give up more of Good B for each new unit of Good A that you produce. This is because some of your resources are equally useful for making Goods A and B while others are much better for one or the other.
For example, imagine that you are running a restaurant and you have lots of cooks in your kitchen. Some of them make desserts and some of them make main dishes. Now let us say that your customers’ tastes change and they want fewer main dishes and more desserts. At first, this is easy to provide. Some of your cooks are pretty good at both kinds of food so you don’t give up that many desserts for each new main dish that you produce. But as you move more and more towards desserts, things get worse. You have some cooks who were really good at making main dishes and could make many of them in a short amount of time, but who are really bad at making desserts and take a long time. When you switch those cooks to desserts, you give up lots of main dishes and only get few desserts in return. Your opportunity costs have risen because you have gotten to the point where your resources (the cooks) are much better at producing one good than the other.
This is why a PPC bows out. In the middle ranges of the PPC, resources are just about equally good for making either thing (PPCs show tradeoffs between making two things or kinds of things), so the opportunity cost for switching is low. However, as you reach the ends of the graph, you get into resources that are only really good for one thing or the other. As you switch those resources, you give up a lot of one product to get only a little bit of the other. In this way, the law of increasing opportunity costs causes a PPC to have a bowed-out shape.