Equal society = happy society?

Does income inequality really matter? Why do politicians like to use words like "class warfare" if not to fuel politics based on greed and envy? In more equal societies, shouldn't it be true that we envy our neighbours less and therefore live happier lives? Given that more equal societies tend to be more equally poor, shouldn't at least this be a consolation prize? Apparently not:

Studies of personal happiness, based on questionnaires and self-reporting, indicate that the inequality of happiness is not growing over time in the United States. Furthermore, the United States has an inequality of happiness roughly comparable to that of Sweden or Denmark, two nations with strongly egalitarian reputations.

Hat Tip: Greg Mankiw