In the coming decades, an increasing competition for global land and water resources can be expected, due to rising demand for food and bio-energy production, biodiversity conservation, and changing production conditions due to climate change. The potential of technological change in agriculture to adapt to these trends is subject to considerable uncertainty. In order to simulate these combined effects in a spatially explicit way, we present a model of agricultural production and its impact on the environment (MAgPIE). MAgPIE is a mathematical programming model covering the most important agricultural crop and livestock production types in 10 economic regions worldwide at a spatial resolution of three by three degrees, i.e., approximately 300 by 300 km at the equator. It takes regional economic conditions as well as spatially explicit data on potential crop yields and land and water constraints into account and derives specific land-use patterns for each grid cell. Shadow prices for binding constraints can be used to valuate resources for which in many places no markets exist, especially irrigation water. In this article, we describe the model structure and validation. We apply the model to possible future scenarios up to 2055 and derive required rates of technological change (i.e., yield increase) in agricultural production in order to meet future food demand.