How does one derive the (asymmetric) demand functions? Traditionally, if one derives the demand function from the (quadratic) utility function U = a1 x1 + a2 x2 + 1/2 (b1 x1^2 + b2 x2^2 + 2 c x1 x2), where x1 and x2 are the quantities,
p1 = dU/dx1 = a1 + b1 x1 + c x2;
p2 = dU/dx2 = a2 + b2 x2 c x1.

What utility function would allow one to obtain asymmetric demand functions of the type

p1 = a1 + b1 x1 + c1 x2;
p2 = a2 + b2 x2 c2 x1.

(note the cross price elasticities c1 anc c2).