When does third-degree price discrimination reduce social welfare, and when does it raise it?

Sufficient conditions are developed for third-degree price discrimination by a monopolist serving all markets to reduce and raise social welfare.  Welfare falls if the demand function in the market whose price is higher with discrimination is at least as convex as that in the other market (at the non-discriminatory price).  Welfare rises if inverse demand in the low-price market is more convex (at the discriminatory price) than inverse demand in the high-price market and the discriminatory prices are close together, so the cost of misallocation is less than the benefit of higher output.