We study how stochasticity in the evolution of agricultural productivity interacts with economic and population growth, and the associated demand for food. We use a two-sector Schumpeterian model of growth, in which a manufacturing sector produces the traditional consumption good and an agricultural sector produces food to sustain contemporary population. In addition, sectors differ in that agriculture also demands land as an input, itself treated as a scarce form of capital. In our model both population and sectoral technological progress are endogenously determined, and key technological parameters of the model are structurally estimated using 1960-2010 data on world GDP, population, cropland and technological progress. Introducing random shocks to the evolution of total factor productivity in agriculture, we show that uncertainty optimally requires more land to be converted into agricultural use as a hedge against production shortages, and that it significantly affects both consumption and population trajectories.