Error and erasure exponents for the asymmetric broadcast channel are analyzed. The focus of the error analysis is on the main receiver where, nominally, both messages are to be decoded. A two-step decoding algorithm is proposed and analyzed. This receiver first attempts to decode both messages, failing which, it attempts to decode only the message representing the coarser information, i.e., the cloud center. This algorithm reflects the intuition that we should decode both messages only if we have confidence in the estimates; otherwise one should only decode the coarser information. The resulting error and erasure exponents, derived using the method of types, are expressed in terms of a new and fundamental quantity-a penalized form of the modified random coding error exponent, reflecting the superposition coding structure of the codebook.