As the previous example demonstrated, utility maximization places no restrictions on the slopes of individual demand functions. However, the consumer's demand must always satisfy the budget constraint, which places certain restrictions on the system of demand functions as a whole. Show that for all goods i,

Engel aggregation condition

Cournot aggregation condition
where ai = pixi/m is the budget share of good i and

are the price and income elasticities of demand respectively.

E1 x = 1