On the existence and uniqueness of first best tolls in networks with multiple user classes and elastic demand

System optimal (SO) or first best pricing is examined in networks with multiple user classes and elastic demand, where different user classes have a different average value of value of time (VOT). Different flows (and first best tolls) are obtained depending on whether the SO characterisation is in units of generalised time or money. The standard first best tolls for time unit system optimum are unsatisfactory, due to the fact that link tolls are differentiated across users. The standard first best tolls for the money unit system optimum may seem to be practicable, but the objective function of the money unit system optimum is nonconvex, leading to possible multiple optima (and non-unique first best tolls). Since these standard first best tolls are unsatisfactory, we look to finding common money tolls which drive user equilibrium flows to time unit SO flows. Such tolls are known to exist in the fixed demand case, but we prove that such tolls do not exist in the elastic demand case. Although common money tolls do not exist which drive the solution to the exact time system optimal flows, tolls do exist which can push the system close to time system optimum (TSO) flows.