Becker's Nobel Prize lecture is titled The Economic Way Of Looking At Life (.pdf) and in it, Becker briefly goes over how he used economics to analyze such topics as crime (why people commit it), discrimination against minorities, and family life.
In his powerful intro, he makes two important points. One is that economists who presume that everyone is selfish (maximizing their own utility) take too narrow a view of human motivation. People are motivated to act for all kinds of reasons (jealousy, spite, etc.). His other point is that while productivity has provided abundance in many areas, the one resource that we'll never have more of his time. Time will always run out, and there will always only be 24 hours in a day. For this reason, we'll never have Utopia, because that scarcity will always be there.