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In a socialist economy the basic economic problems are solved through planning.
Without scarcity, the study of economics would not exist because we would produce, distribute and consume everything to meet all of our needs.
There would be no choice about consumption because it would be unlimited.
A PPF is a graph showing the possible production combinations of two goods given limited inputs. Society can produce two goods - gadgets or widgets.
The downward sloping line shows how many gadgets and widgets it can produce given the amount of labor available.
However, that combination is unattainable given the limited resource of labor. In summation, scarcity exists because we live in a society with limited resources and unlimited wants and needs.
It forces us as a society and individual people to make decisions every day.
Why are certain decisions made and what is the next best alternative that was forgone?
As noted, if scarcity did not exist, all goods and services would be free.
As you increase production of gadgets, you have to decrease production of widgets, and vice versa. Assume the unlimited demand of gadgets was 100 and widgets was 300. All points along and inside the PPF are attainable.
If labor was not scarce, society would produce 100 gadgets and 300 widgets. Because society cannot meet the unlimited demand given the limited amount of labor, gadgets and widgets are considered scarce.