A linear programming food price index permits the possibility of routine substitutions in response to price changes, while a fixed weight index does not. This paper cites a variety of evidence indicating that consumers do trade off among foods in response to both long-term and seasonal price changes. Implications of this for food price indexes are discussed. A linear programming food price index is then developed, based on an empirical menu-planning model, and compared with corresponding fixed weight price indexes over a nine-month period. The linear programming index shows a significantly smaller increase in food prices over the nine-month period than the fixed weight index with initial period weights.