The purpose of this work is to find empirical evidence on the behavior of the prices in an insular region, with a market of small size and, simultaneously, fragmented. To that end, with consumer price data for four products of the food and beverage group observed in the Canary Islands we study the price and rigidity, characterized by frequency, duration and size of price changes. The main empirical result found for these four products is that market fragmentation and the type of establishment are important factors in explaining the price level and not so when one wants to explain his change. In this case, the product is the decisive element in explaining the basic features of price change, not observing rigidities either in terms of changes or establishments.