In many cases, heterogeneous prices occur in price-taking environments. In empirical work, an aggregate price index is often created from such prices and used to explain aggregate supply and derived demands. The authors show that consistent aggregation places strong restrictions on functional...

"Crop production is subject to supply shocks, and both expected and realized outputs as well as output prices are unknown when inputs are chosen. The process by which producers form expectations is difficult to model, especially when working with aggregate data. We present a necessary and...

Duality methods for incomplete systems of consumer demand equations are adapted to the dual structure of variable cost functions in joint production. This allows identification of necessary and sufficient restrictions on technology and cost so that conditional factor demands can be written as...