Climate anomalies pose risks to agriculture and food security. To assess the impact, this paper models the complex dependences of climate extreme indices and the crop-related variables: yield, production, and price of a crop. Using a comprehensive copula-based analysis, the conditional distributions of the crop-related variables given extremes of air temperature and precipitation are estimated. We used potatoes in the Netherlands as a case study. Weather data were obtained from 33 weather stations and ECMWF ERA-interim archive during the period 1980–2017. A joint behavior analysis predicted the yield, the production and the price with the relative mean absolute error equal to 5.4%, 3.6%, and 27.9%, respectively. The study showed that copulas adequately describe the multivariate dependences. Those in turn are able to quantify the impact of climate extremes, including their uncertainties.