2. Aggregation across goods Let e(p, v) be the expenditure function, then the problem is max(v(x), z) v(x),Z s.t. e(p, v)+qz=w If e(p, v)is liner as e(p)y(v is homothetic),let P=e(p)and X=v(x), then max(x, z) X s.t. PX +qz=w2.Aggregation across goods • Let be the expenditure function, then the problem is: • If is liner as (v is homothetic), let and ,then e p v ( , ) ( ) max ( ( ), ) . . ( , ) v x U v s t e p v w + = , z x z qz e p v ( , ) e p v ( ) P e p = ( ) X v = ( ) x max ( , ) . . X U X s t PX w + = , z z qz