We present a revealed preference methodology for nonparametric demand analysis under the assumption of normal goods. Our methodology is flexible in that it allows for imposing normality on any subset of goods. We show the usefulness of our methodology for empirical welfare analysis through cost-of-living indices. An illustration to US consumption data drawn from the Panel Study of Income Dynamics (PSID) demonstrates that mild normality assumptions can substantially strengthen the empirical analysis. It obtains considerably tighter bounds on cost-of-living indices and a significantly more informative classification of better-off and worse-off individuals after the 2008 financial crisis.