One important source of uncertainty is the response of the climate to change in GHG atmospheric concentration, as well as the resulting patterns of change in terms of precipitation and temperature changes. We used in this paper results from five GCMs but a larger set of results is produced by the CMIP5 community (see full list and results overview in (Collins _et al._ , 2013)). Uncertainties for projections are large in these models, related to various processes difficult to model and whose net effect remain largely uncertain: e.g., the dynamics of cloudiness (in particular in the tropics), the role of surface albedo, or climate induced changes in carbon fluxes (Andrews _et al._ , 2012; Vial _et al._ , 2013; Friedlingstein _et al._ , 2014). As an illustration, although the projected changes in global mean temperature are much less uncertain compared to regional patterns or precipitation changes, CMIP5 models estimate such a global temperature increase to range between 2.1 and 4.7 °C (Andrews _et al._ , 2012) for a given degree of radiative forcing (quadrupling). As detailed in section 3.2, our climate change scenarios were specifically selected to span a large set of climate responses, and this source of uncertainty should be kept in mind when looking at the average results on a RCP.

The second source of uncertainty stems from the modeling of the crop responses themselves. For this paper, we relied on only one biophysical model, EPIC. However, as illustrated in Rosenzweig _et al._ (2014), the most commonly used biophysical models predict very different responses to change in temperature and precipitation under high global warming. A pattern commonly observed in crop impact research is a small increase in crop yields in high and middle latitudes under low to moderate global warming and negative impact in tropical areas. When large ensemble of models are compared, impacts by 2050 on major crops remain within the range ‐25%‐+10% for four fifths of the projections (Porter _et al._ , 2014). However, when looking at details per crop and management type (low or high input, irrigation), results can vary considerably depending on model parameterization (Rosenzweig _et al._ , 2014). One important parameter for instance appears to be the modeling of nitrogen stress and the consideration of historical fertilization patterns. To this extent, our modeling framework explicitly takes into account the diversity of management systems and their specific response to climate change (Leclère _et al._ , 2014).

Another important source of uncertainty at the crop level is the influence of elevated atmospheric CO2 concentration. According to the IPCC, when CO2 effects are taken into account, crop prices are “as likely as not”, with range of expected responses from ‐30% to + 45% (Porter _et al._ , 2014). Recent studies illustrate that CO2 effects are not only likely to buffer a part of the impact for crops, but also to boost grassland productivity, with important implications for the livestock sector. Havlík _et al._ (2015) report a global average effect of climate change of +12.2% on grass yield globally using the EPIC model, and +45% with the LPJmL model for a RCP8.5 scenario with CO2 effects by 2050. Without CO2 effects, these impacts would be +2% and +7%, respectively. However, as emphasized by Rosenzweig et al., dispersion across model results are much higher with CO2 effects than without, and substantial additional efforts should be devoted to CO2 effects modeling. Additionally, there is some indication that crop quality, in terms of nutrient content, may decrease under CO2 effects, which would moderate the expected benefits of yield increases (Myers _et al._ , 2014). For these two reasons, the RCP8.5* scenario that does not include the effects of elevated CO2 concentration is an important counterfactual to consider in this paper, based here on the HadGEM2‐ES model.

There are however some other sources of yield impacts that are absent or imperfectly represented in our modeling framework and could lead to either higher or lower climate change biophysical impacts. Some management responses are partially reflected in a crop model such as EPIC (adjustment of 38