Many previous studies have evaluated the hydrologic response to climate change using the first‐order approximation (first‐order Taylor expansion) of the Mezentsev‐Choudhury‐Yang equation (formulating the Budyko hypothesis), which has a parameter n representing catchment characteristics. However, no studies have paid attention to the error due to the first‐order approximation. This study therefore estimates this error to improve the theoretical framework for assessing the contribution of climate change to runoff based on the Budyko hypothesis. Specifically, the error increases when precipitation (P) decreases and potential evaporation (E0) increases, and n increases. Therefore, an increasing P or decreasing E0 leads to an underestimate of the climatic contributions, while a decreasing P or increasing E0 leads to an overestimate. In addition, we suggest a new method to accurately estimate the contribution of climate change to runoff. Key Points: Error analysis of the first‐order Taylor expansion of Budyko hypothesis A new method for assessing hydrologic response to climate change