a. Constant Opportunity Costs 

A constant opportunity cost implies that the trade-off between producing one more unit of a good and the amount of the other good that must be given up remains the same, regardless of the initial levels of production.

In other words, if the production possibilities frontier (PPF) were linear, the opportunity cost of producing additional units of one good would be fixed.

However, in the case of a bowed-out (concave) PPF, this is not the situation. Here, the curvature of the PPF indicates that resources are specialized and not equally efficient in producing both goods.

As production of one good increases, the resources that are most adept at producing that good become scarcer, and resources that are better suited for producing the other good are utilized. This leads to an increasing opportunity cost, meaning that to produce each additional unit of the chosen good, more and more of the other good must be foregone.

b. Decreasing Opportunity Costs 

Decreasing opportunity costs would imply a convex (bowed-in) PPF. This means that as an economy shifts resources from producing one good to the other, the trade-off becomes more favorable.

In essence, as you produce more of one good, the resources that are increasingly specialized for producing that good are freed up, making it easier (less costly) to produce additional units.

However, a bowed-out PPF implies the opposite. Initially, resources used to produce one good are relatively versatile, but as more of that good is produced, the resources that are better suited for producing the other good are depleted.

This leads to an increasing opportunity cost, meaning that to produce more of one good, an increasing amount of the other good must be sacrificed.

c. First Increasing and then Decreasing Opportunity Costs 

This scenario would imply a highly unusual and non-standard PPF. It would suggest that initially, as an economy produces more of one good, the opportunity cost increases (as with a bowed-out PPF), but then at some point, it starts to decrease.

This would be a highly complex and contradictory representation of production possibilities.

Such a situation doesn’t align with the fundamental economic concept of opportunity cost. The opportunity cost typically increases as you move along the PPF, reflecting the inherent trade-offs involved in resource allocation.

In conclusion, a bowed-out PPF signifies increasing opportunity costs, indicating that as a country produces more of one good, it must increasingly sacrifice the production of the other good. This reflects the economic reality of scarcity and the necessity of making trade-offs in resource allocation.