Grammar-based compression is a loss-less data compression scheme that represents a given string w by a context-free grammar that generates only w. While computing the smallest grammar which generates a given string w is NP-hard in general, a number of polynomial-time grammar-based compressors which work well in practice have been proposed. RePair, proposed by Larsson and Moffat in 1999, is a grammar-based compressor which recursively replaces all possible occurrences of a most frequently occurring bigrams in the string. Since there can be multiple choices of the most frequent bigrams to replace, different implementations of RePair can result in different grammars. In this paper, we show that the smallest grammars generating the Fibonacci words Fk can be completely characterized by RePair, where Fk denotes the k-th Fibonacci word. Namely, all grammars for Fk generated by any implementation of RePair are the smallest grammars for Fk, and no other grammars can be the smallest for Fk. To the best of our knowledge, Fibonacci words are the first non-trivial infinite family of strings for which RePair is optimal.