Arthur Brooks is an economist, so it is not surprising that the latter half of Gross National Happiness is devoted to the correlation of different economic conditions with happiness.
His main finding is that happiness correlates most with your view of mobility, not with actual inequality.
The United States has a high level of income inequality compared to other industrialized nations. However, how happy people are does not correlate well with what income they have. Instead, the happiest people are the ones who believe that you can change your income if you want to - regardless of what your income is now or of how much it has actually changed. Likewise, the least happy people are the ones who believe that it is hard to change your income - regardless of what your income is now or how much it has actually changed.
Brooks is a free marketeer, though not an all-out libertarian (he is, after all, a practicing Catholic). He thinks that free markets promote mobility, and they promote the belief in mobility. Both of these things -- actual mobility, and the belief in mobility -- add to gross national happiness.
Mrs. G. thinks Brooks should have made a much stronger case for the Catholic social teaching that promotes property ownership for all. I agree, though I think this is more a difference of emphasis in Brooks' argument than of his actually rejecting Catholic social teaching.
The summary number that Brooks gives us is a comparison of two measures of inequality (the Gini coefficient). On a scale from 0 to 1, with 0 meaning everyone is the same and 1 means they are all different, the income inequality measure for the United States is .44, which is fairly high. On the other hand, when we compare how people think about mobility, the difference among Americans is only .18. The gloomy end of the distribution, who think that it is hard to change class position, is a pretty small -- and unhappy - minority.